You're going to run into a bunch of problems:
First and most obvious: Your beam fire control tracking speed is way too slow. While the tracking speed of your weapon is fine (being the same as your ship speed), for trying to hit a target the BFC tracking speed also matters and the lower of the two is going to be your accuracy limiting factor. Your high speed is completely wasted here.
A second thing, while a lot of players prefer to build very small fighters usually for RP reasons, very rarely is there any good reason not to build fighters as close to the 500-ton limit as possible. Fighters suffer the most from tonnage efficiency losses due to small size, so minimizing those losses by building a larger ship is usually the best approach. If it is for RP reasons then it is what it is, however a smaller fighter is not the best way to maximize speed - speed is a function of the ratio between engine and total mass/displacement, not a function of ship mass only. You can in this case nearly quadruple the engine size, build a 500-ton fighter, and get the same speed with large efficiency gains in terms of firepower.
Gauss cannons are usually a poor weapon choice for fighters until fairly high techs, as a 10 cm railgun is usually more tonnage-efficient. The reason is that Gauss cannons are most effective as turreted weapons, which allows them to reach your maximum tracking speed (4x racial tech level) while railguns are limited by the speed of the ship they are mounted on. For a fast, fast fighter this limitation for a railgun is very minimal so a turreted Gauss cannon has comparatively little advantage for the extra tonnage spent. Here you have not turreted the Gauss cannon so it is simply not effective, for 0.5 HS (25 tons) you are dealing 0.24 expected damage per 5-second increment (0.48 per HS). A three-shot 10 cm railgun can deal 3 damage per increment on a 2.325 HS gun (1.29 per HS), while a full four-shot railgun (3 HS) can deal 1.33 per HS. Even with the added tonnage of a R3 gas-cooled reactor (0.72 HS) this is much higher damage potential. Gauss cannons only match or exceed railguns for fighters once you reach Rate of Fire 6 tech level.
Finally, including both missile and beam on the same fighter is sub-optimal. You would get more efficiency from having two classes, one with twice as many guns and one with twice as many launchers, reducing the net tonnage needed for fire controls. In general I dislike putting AMMs on fighters, as they don't benefit from the fighter as a weapons platform. Usually it is better to have a larger AMM ship with a big RES-1 sensor and MFC to provide rapid AMM fire with fast reloading and deep magazines. However if you want it for RP reasons, more power to you - but it would be better to put the AMMs on a different class of fighter.
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If I take into account the above comments, but keep the dual gun/missile armament for flavor purposes, using the maximum 500 tons something like this is possible:
F-1 Heavy class Interceptor (P) 500 tons 20 Crew 98.5 BP TCS 10 TH 120 EM 0
12023 km/s Armour 1-5 Shields 0-0 HTK 3 Sensors 0/0/0/0 DCR 0 PPV 2.55
Maint Life 0 Years MSP 0 AFR 99% IFR 1.4% 1YR 24 5YR 357 Max Repair 60.00 MSP
Magazine 6
Lieutenant Commander Control Rating 1
Intended Deployment Time: 0.6 days Morale Check Required
Ion Drive EP120.00 (1) Power 120.0 Fuel Use 571.55% Signature 120.00 Explosion 20%
Fuel Capacity 9,000 Litres Range 0.57 billion km (13 hours at full power)
10cm Railgun V30/C1.5/S2 (1x2) Range 30,000km TS: 12,023 km/s Power 1.5-1.5 RM 30,000 km ROF 5
Beam Fire Control R48-TS12000 (SW) (1) Max Range: 48,000 km TS: 12,000 km/s 79 58 38 17 0 0 0 0 0 0
Gas-Cooled Fast Reactor R1.5 (1) Total Power Output 1.5 Exp 5%
Size 1 Box Launcher (6) Missile Size: 1 Hangar Reload 50 minutes MF Reload 8 hours
Missile Fire Control FC3-R1 (1) Range 3.5m km Resolution 1
This design is classed as a Fighter for production, combat and planetary interaction
This design is classed as a e for auto-assignment purposes
Given the need for a bit more tonnage to fix the BFC range I settled for basically tripling the smaller design in terms of firepower. The two-shot railgun plus power cell is 105 tons, which is about the same as four 0.25-HS Gauss guns but puts out twice the DPS. With your current tech level there is no significant benefit to going above 12,000 km/s as you can't get a higher BFC speed, which puts an upper limit on how much engine power is needed. If you researched more BFC speed techs then more speed would be useful.