If you are going to rely on weapon doctrines to distinguish your factions I think you are going to have a lot of difficulty accomplishing this if you reserve missiles as something all three factions use. Three factions with beam races are frankly going to be fairly similar as the differences between beams are not nearly as large as the difference between beam and missile combat.
You might alternatively consider basing the three factions around the three weapon types that can each be used as a single-weapon fleet basis, which are Railguns, Lasers, and Missiles. All of these are capable of filling anti-ship and PD roles to a serviceable extent but in such a way that you get a "soft" weapons triangle with missiles > lasers > railguns > missiles - as laser turrets are the worst PD, but beat railguns at range unless the railgun fleet has high enough speed, while railgun PD is quite strong especially at low tech levels before Gauss becomes stronger.
Then each faction could use a complementary secondary weapon, e.g. the Laser Faction adds Gauss for better PD, the Railgun Faction adds PBs to cover the long range niche, and the Missile Faction adds HPMs to blind enemies who close in to facilitate their escape. Of course not every weapon type must be used necessarily.
That said, even so a potential problem with limiting weapon choices is that each faction is very limited in how they could evolve their doctrines over time in response to events. This may be acceptable if you're not heavy on RPing and just want to make three different fleets fight each other but is worth noting. For me this is a big reason among others that I would prefer to differentiate factions by other factors and leave the weapon choices more open-ended.
Regarding your specific combinations:
- Railguns and Plasma don't work well together. They are both short-range weapons and as railguns are quite strong aside from lacking in penetration there is not much that plasma contributes. Railguns are an ideal weapon to combine with a specialized secondary weapon instead as the railguns are so all-purpose.
- To combine with lasers I think the best choice is something that is specialized, but not in that it has weird mechanics like mesons or HPMs, rather a weapon type that is very good at a specific niche. Gauss is ideal for this as it excels at PD which lasers are a bit weak in despite having access to turrets. Mesons I think are generally just too weak to be useful.
- PBs and Gauss are a strong combo, but research-intensive. The trouble with PBs is they basically lock you into railguns or Gauss as otherwise you have no good PD, which is why I suggest using missile as the third primary weapon.
It might also be good to think beyond just weapons and ask more questions about how each faction designs their fleet. There are many options here which are even weapon-agnostic, e.g. you can pursue a carrier strike doctrine, capital ships, cruiser fleet, FAC and other smaller ships, even exotic ideas like the tractor beam "block ship" idea Zap0 has used in their fiction universe. Similarly, in terms of design you can have varying philosophies on how to use the tonnage of your ship, e.g. one faction may have more and larger engines, another may go for heavy armor, a third might spend the RPs to develop strong shields and/or cloaking devices. Not only can these effectively differentiate a faction but in fact they can guide weapon choices in a quite organic manner.
I am using a Railgun Gauss race currently. Railguns are the main guns and Gauss are for Pd.
the interesting thing about this combination is that they do not require any Power Plants leaving more space for payload amending the lesser ductility of the kinetic weapons.
They use variegated missile ranges depending on the situation, if they cannot close in they will use mid to low-range if they are while if they can close in they use Long-range to start weakening the defenses. Because of that, each ship is classified as A, B, or C where:
A has long-range (afar)
B has both (both)
C has short-range (close)
Missiles are usually all size 10 so the bulk ship is fundamentally the same and all is changing are Sensors and Ordnance helping with retooling and production.
Railguns totally require power plants...