This was a multi-role design that I never had a chance to field (game devolved into 30 second intervals).
New Orleans class Heavy Cruiser    30,000 tons     2808 Crew     10600.6 BP      TCS 600  TH 990  EM 0
6875 km/s     Armour 14-86     Shields 0-0     Sensors 70/70/0/0     Damage Control Rating 120     PPV 125.52
Maint Life 14.48 Years     MSP 15457    AFR 102%    IFR 1.4%    1YR 138    5YR 2066    Max Repair 600 MSP
Magazine 200   

MCFD E7.2 (30)    Power 137.5    Fuel Use 72%    Signature 33    Armour 0    Exp 10%
Fuel Capacity 2,485,000 Litres    Range 207.1 billion km   (348 days at full power)

Twin 15cm C5 Ultraviolet Laser Turret (4x2)    Range 240,000km     TS: 25000 km/s     Power 12-10     RM 4    ROF 10        6 6 6 6 4 4 3 3 2 2
Triple 20cm C5 Ultraviolet Laser Turret (2x3)    Range 320,000km     TS: 13900 km/s     Power 30-15     RM 4    ROF 10        10 10 10 10 8 6 5 5 4 4
AMFC S16 160-25000 (2)    Max Range: 320,000 km   TS: 25000 km/s     97 94 91 88 84 81 78 75 72 69
ASFC S12 160-15000 (1)    Max Range: 320,000 km   TS: 15000 km/s     97 94 91 88 84 81 78 75 72 69
ICFR PB-1.1 (6)     Total Power Output 79.2    Armour 0    Exp 10%

Standard Box Launcher (50)    Missile Size 4    Hangar Reload 30 minutes    MF Reload 5 hours
MFC151-R100 (1)     Range 151.2m km    Resolution 100
Striker SM Mk. 1 (50)  Speed: 27,500 km/s   End: 63.6m    Range: 105m km   WH: 8    Size: 4    TH: 165 / 99 / 49

Standard Anti-Missile Sensor MR50-R1 (1)     GPS 360     Range 50.4m km    Resolution 1
Standard Search Sensor MR252-R100 (1)     GPS 18000     Range 252.0m km    Resolution 100
Emergency Search Sensor MR50-R100 (1)     GPS 3600     Range 50.4m km    Resolution 100
Thermal Sensor TH5-70 (1)     Sensitivity 70     Detect Sig Strength 1000:  70m km
EM Detection Sensor EM5-70 (1)     Sensitivity 70     Detect Sig Strength 1000:  70m km
I am pretty adamant about having long maintenance lives. These ships are intended to operate with a command/jump ship that also has two of the twin 15cms for additional PD volume, as the "battle line" of fleets consisting of 5 New Orleans, 1 Command Cruiser, 5 Destroyers (Missile ships), 1 Destroyer Leader, 5 Destroyer Escorts (AMM ships), and 1 Escort Leader.

Depends on your logistics situation, do you have enough colliers to keep a fleet supplied for multiple engagements? Do you have the industry to keep up with usage?

I think you will find that the AMM ships will be more effective at stopping missiles, but you will burn through them like nobodies business in a fight, so if you can keep them supplied then those would be a good focus. If you aren't certain you can, then more gauss would be better. Like, lots more.

The new designs appear much more capable, I would be interested in hearing how they perform.

Well, I haven't actually found anything to kill yet( ), so that might be a while.
Oops!  I had the ordnance numbers from the double sized Majestic there, it should have 250 Serpentine Mk. IIIs and 500 Gargoyle Mk. IIs. I also forgot to update a few ships to the Serpentine Mk. III (The one with thermal sensors(Can they detect ships right on top of them?)), that should be what's in the Fenrir and Majestic's magazines.

You forgot the civilian habitat and cargo hold.

Somewhere in your Imperium, a designer is very pleased with himself, and an engineer is committing suicide.

Alright, updated my designs and added a few new ones:

First, new missiles (I've added agility to both(And a thermal sensor to the ASM)):

Missile Size: 4 MSP  (0.2 HS)     Warhead: 9    Armour: 0     Manoeuvre Rating: 14
Speed: 47000 km/s    Endurance: 16 minutes   Range: 45.0m km
Thermal Sensor Strength: 0.12    Detect Sig Strength 1000:  120,000 km
Cost Per Missile: 5.6033
Chance to Hit: 1k km/s 658%   3k km/s 210%   5k km/s 131.6%   10k km/s 65.8%
Materials Required:    2.25x Tritanium   0.12x Uridium   3.2583x Gallicite   Fuel x1250

Development Cost for Project: 560RPMissile Size: 1 MSP  (0.05 HS)     Warhead: 1    Armour: 0     Manoeuvre Rating: 60
Speed: 31400 km/s    Endurance: 1 minutes   Range: 2.5m km
Cost Per Missile: 2.024
Chance to Hit: 1k km/s 1884%   3k km/s 600%   5k km/s 376.8%   10k km/s 188.4%
Materials Required:    0.25x Tritanium   1.524x Gallicite   Fuel x17.5

Development Cost for Project: 202RP
And now, the ships:

First off, my command frigate, with boosted versions of the entire fleets sensor capability. It can provide long range sensor coverage for the entire fleet, but all of my ships are still fitted with appropriate individual sensors for their weapons only just in case it is destroyed. It has additional armor and shields compared to the rest of the fleet, as it will likely be targeted first off, and carries a few SWACs and a jump scout.
Spectre class Frigate Leader    10,000 tons     744 Crew     3426 BP      TCS 200  TH 230.4  EM 1200
4800 km/s     Armour 8-41     Shields 40-300     Sensors 240/240/0/0     Damage Control Rating 36     PPV 0
Maint Life 2.59 Years     MSP 1178    AFR 145%    IFR 2%    1YR 247    5YR 3703    Max Repair 360 MSP
Hangar Deck Capacity 1000 tons    

ME5 Insulated MPD Engine (12)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 650,000 Litres    Range 234.0 billion km   (564 days at full power)
R4E5 Theta Shield Projector (10)   Total Fuel Cost  200 Litres per day

S36/24-100 Battle Management System (1)     GPS 36000     Range 864.0m km    Resolution 100
S36/24-16 Missile Attack Control Sensor (1)     GPS 5760     Range 345.6m km    Resolution 16
S36/24-1 Point Defense Control Sensor (1)     GPS 360     Range 86.4m km    Resolution 1
S24-240 Thermal Sensor Array (1)     Sensitivity 240     Detect Sig Strength 1000:  240m km
S24-240 EM Sensor Array (1)     Sensitivity 240     Detect Sig Strength 1000:  240m km

Strike Group
1x Vanguard Jump Scout   Speed: 9600 km/s    Size: 5
3x Bloodhound AWAC   Speed: 9600 km/s    Size: 5

This design is classed as a Military Vessel for maintenance purposes
Now for the line ships; the Fenrir is now equipped with additional magazine space and has lost its beam array:
Fenrir class Missile Frigate    10,000 tons     932 Crew     2304.4 BP      TCS 200  TH 230.4  EM 600
4800 km/s     Armour 6-41     Shields 20-300     Sensors 1/1/0/0     Damage Control Rating 35     PPV 40
Maint Life 3.85 Years     MSP 720    AFR 160%    IFR 2.2%    1YR 77    5YR 1149    Max Repair 80 MSP
Magazine 940    

ME5 Insulated MPD Engine (12)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 500,000 Litres    Range 180.0 billion km   (434 days at full power)
R4E5 Theta Shield Projector (5)   Total Fuel Cost  100 Litres per day

SS4 Missile Launch Tube (10)    Missile Size 4    Rate of Fire 20
S36/24-16 Missile Fire Control (2)     Range 82.9m km    Resolution 16
Serpentine Mk. II (235)  Speed: 49,000 km/s   End: 15.3m    Range: 45m km   WH: 9    Size: 4    TH: 228 / 137 / 68

S36/24-16 Active Grav Sensor Array (1)     GPS 922     Range 55.3m km    Resolution 16

Missile to hit chances are vs targets moving at 3000 km/s, 5000 km/s and 10,000 km/s

This design is classed as a Military Vessel for maintenance purposes
For point defense, I've refitted the Scipio with many more fast Gauss guns, and removed all of the heavy ones:
Scipio class Escort Frigate    10,000 tons     757 Crew     3639.4 BP      TCS 200  TH 230.4  EM 840
4800 km/s     Armour 6-41     Shields 28-300     Sensors 1/1/0/0     Damage Control Rating 35     PPV 44
Maint Life 5.9 Years     MSP 3137    AFR 160%    IFR 2.2%    1YR 154    5YR 2304    Max Repair 270 MSP

ME5 Insulated MPD Engine (12)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 500,000 Litres    Range 180.0 billion km   (434 days at full power)
R4E5 Theta Shield Projector (7)   Total Fuel Cost  140 Litres per day

Triple Fast Defensive Gauss Turret (11x15)    Range 50,000km     TS: 28700 km/s     Power 0-0     RM 5    ROF 5        1 1 1 1 1 0 0 0 0 0
72000-25000 Point Defence Beam Fire Control (6)    Max Range: 144,000 km   TS: 25000 km/s     93 86 79 72 65 58 51 44 37 31

S36/24 Beam PD Sensor (1)     GPS 4     Range 860k km    Resolution 1

This design is classed as a Military Vessel for maintenance purposes
The Loki has recieved more magazine space but is largely unchanged:
Loki class Escort Frigate    10,000 tons     712 Crew     2314.9 BP      TCS 200  TH 230.4  EM 600
4800 km/s     Armour 6-41     Shields 20-300     Sensors 1/1/0/0     Damage Control Rating 35     PPV 12
Maint Life 3.51 Years     MSP 760    AFR 152%    IFR 2.1%    1YR 95    5YR 1419    Max Repair 99 MSP
Magazine 1362    

ME5 Insulated MPD Engine (12)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 500,000 Litres    Range 180.0 billion km   (434 days at full power)
R4E5 Theta Shield Projector (5)   Total Fuel Cost  100 Litres per day

Size 1 Missile Launcher (12)    Missile Size 1    Rate of Fire 5
S36/24-1 Missile Fire Control (4)     Range 31.1m km    Resolution 1
Gargoyle Mk. II (1287)  Speed: 31,400 km/s   End: 1.3m    Range: 2.5m km   WH: 1    Size: 1    TH: 628 / 376 / 188

S36/24-1 Active Grav Sensor (1)     GPS 99     Range 23.8m km    Resolution 1

Missile to hit chances are vs targets moving at 3000 km/s, 5000 km/s and 10,000 km/s

This design is classed as a Military Vessel for maintenance purposes

The Jester is a dedicated anti-ship beam frigate, with slightly higher speed for chasing down enemy ships and heavier shields:
Jester class Frigate    10,000 tons     1150 Crew     3609.4 BP      TCS 200  TH 249.6  EM 1560
5200 km/s     Armour 6-41     Shields 52-300     Sensors 1/1/0/0     Damage Control Rating 35     PPV 54
Maint Life 3.81 Years     MSP 1128    AFR 160%    IFR 2.2%    1YR 122    5YR 1829    Max Repair 281 MSP

ME5 Insulated MPD Engine (13)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 500,000 Litres    Range 180.0 billion km   (400 days at full power)
R4E5 Theta Shield Projector (13)   Total Fuel Cost  260 Litres per day

F20C5 Far Ultraviolet Laser Projector (9)    Range 500,000km     TS: 6250 km/s     Power 10-5     RM 5    ROF 10        10 10 10 10 10 8 7 6 5 5
300000-6250 Long Range Beam Fire Control (3)    Max Range: 600,000 km   TS: 6250 km/s     98 97 95 93 92 90 88 87 85 83
Stellarator Fusion Reactor Technology PB-1 (2)     Total Power Output 36    Armour 0    Exp 5%
Stellarator Fusion Reactor Technology PB-1 (3)     Total Power Output 9    Armour 0    Exp 5%

S36/24-1 Active Grav Sensor (1)     GPS 99     Range 23.8m km    Resolution 1

This design is classed as a Military Vessel for maintenance purposes
The Jump Scout and AWAC carried by my Frigate Leaders:
Vanguard class Jump Scout    250 tons     7 Crew     113.6 BP      TCS 5  TH 11.52  EM 0
9600 km/s    JR 1-50     Armour 1-3     Shields 0-0     Sensors 1/1/0/0     Damage Control Rating 0     PPV 0
Maint Life 9.07 Years     MSP 28    AFR 5%    IFR 0.1%    1YR 1    5YR 9    Max Repair 36 MSP

E8 Fighter Jump Drive     Max Ship Size 400 tons    Distance 50k km     Squadron Size 1
FE5 Insulated MPD Engine (1)    Power 48    Fuel Use 5000%    Signature 11.52    Armour 0    Exp 25%
Fuel Capacity 20,000 Litres    Range 2.9 billion km   (3 days at full power)

S36/24 Near Target Detection System (1)     GPS 576     Range 34.6m km    Resolution 16
S36/24 Near Missile Detection System (1)     GPS 36     Range 8.6m km    Resolution 1

This design is classed as a Fighter for production, combat and maintenance purposes
Bloodhound class AWAC    250 tons     6 Crew     122.6 BP      TCS 5  TH 11.52  EM 0
9600 km/s     Armour 1-3     Shields 0-0     Sensors 1/1/0/0     Damage Control Rating 0     PPV 0
Maint Life 0 Years     MSP 0    AFR 50%    IFR 0.7%    1YR 12    5YR 180    Max Repair 50 MSP

FE5 Insulated MPD Engine (1)    Power 48    Fuel Use 5000%    Signature 11.52    Armour 0    Exp 25%
Fuel Capacity 55,000 Litres    Range 7.9 billion km   (9 days at full power)

S36/24 Fighter Target Detection System (1)     GPS 807     Range 48.4m km    Resolution 16
S36/24 Near Missile Detection System (1)     GPS 36     Range 8.6m km    Resolution 1

This design is classed as a Fighter for production, combat and maintenance purposes

And the support vessels:
Kilimanjaro class Oiler    10,000 tons     694 Crew     2408 BP      TCS 200  TH 230.4  EM 600
4800 km/s     Armour 6-41     Shields 20-300     Sensors 1/1/0/0     Damage Control Rating 6     PPV 0
Maint Life 6.53 Years     MSP 903    AFR 133%    IFR 1.9%    1YR 37    5YR 548    Max Repair 80 MSP

ME5 Insulated MPD Engine (12)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 5,000,000 Litres    Range 1800.0 billion km   (4340 days at full power)
R4E5 Theta Shield Projector (5)   Total Fuel Cost  100 Litres per day

This design is classed as a Military Vessel for maintenance purposesMajestic class Collier    10,000 tons     587 Crew     2143 BP      TCS 200  TH 230.4  EM 600
4800 km/s     Armour 6-41     Shields 20-300     Sensors 240/1/0/0     Damage Control Rating 5     PPV 0
Maint Life 3.7 Years     MSP 670    AFR 160%    IFR 2.2%    1YR 76    5YR 1142    Max Repair 240 MSP
Magazine 1500    

ME5 Insulated MPD Engine (12)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 550,000 Litres    Range 198.0 billion km   (477 days at full power)
R4E5 Theta Shield Projector (5)   Total Fuel Cost  100 Litres per day

Serpentine Mk. II (375)  Speed: 49,000 km/s   End: 15.3m    Range: 45m km   WH: 9    Size: 4    TH: 228 / 137 / 68
Gargoyle Mk. II (1509)  Speed: 31,400 km/s   End: 1.3m    Range: 2.5m km   WH: 1    Size: 1    TH: 628 / 376 / 188

S24-240 Thermal Sensor Array (1)     Sensitivity 240     Detect Sig Strength 1000:  240m km

Missile to hit chances are vs targets moving at 3000 km/s, 5000 km/s and 10,000 km/s

This design is classed as a Military Vessel for maintenance purposes
Shield-Bearer class Supply Ship    10,000 tons     510 Crew     1778 BP      TCS 200  TH 230.4  EM 600
4800 km/s     Armour 6-41     Shields 20-300     Sensors 1/1/0/0     Damage Control Rating 5     PPV 0
Maint Life 17.77 Years     MSP 18556    AFR 160%    IFR 2.2%    1YR 111    5YR 1668    Max Repair 80 MSP

ME5 Insulated MPD Engine (12)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 500,000 Litres    Range 180.0 billion km   (434 days at full power)
R4E5 Theta Shield Projector (5)   Total Fuel Cost  100 Litres per day

This design is classed as a Military Vessel for maintenance purposes
Achilles class Jump Frigate    10,000 tons     846 Crew     2708 BP      TCS 200  TH 230.4  EM 2040
4800 km/s    JR 6-500     Armour 10-41     Shields 68-300     Sensors 1/1/0/0     Damage Control Rating 161     PPV 0
Maint Life 7.72 Years     MSP 2862    AFR 72%    IFR 1%    1YR 85    5YR 1273    Max Repair 342 MSP

MS10kS6R500M Jump Drive     Max Ship Size 10000 tons    Distance 500k km     Squadron Size 6
ME5 Insulated MPD Engine (12)    Power 80    Fuel Use 50%    Signature 19.2    Armour 0    Exp 5%
Fuel Capacity 750,000 Litres    Range 270.0 billion km   (651 days at full power)
R4E5 Theta Shield Projector (17)   Total Fuel Cost  340 Litres per day

This design is classed as a Military Vessel for maintenance purposes
These are designed to operate in squadrons with the following layout:

5x Fenrir class Missile Frigate
5x Loki class Escort Frigate
3x Scipio class Escort Frigate
2x Spectre class Frigate Leader
1x Kilimanjaro class Oiler
1x Shield-Bearer class Supply Ship
3x Majestic class Collier
4x Achilles class Jump Frigate
(5 Fenrir and 1 Achilles; 5 Loki and 1 Achilles; 3 Scipio and 2 Spectre; 1 Kilimanjaro and 1 Shield-Bearer and 3 Majestic and 1 Achilles.)

Could you name one advantage of combining ASM and AS Beam ships into one multirole design that would not apply to the same weapons being present in two different ships?

The primary one would be more ships under construction at once, as you only need one slipway rather than the two for seperate ships. Of course this only applies if you have a sufficiently large shipyard to build the multirole ship. For a combined ASM/beam ship, designing it primarily as a beam ship with a stack of box launchers strapped to it to give a big alpha strike capability can prove effective.