I would like to see three new ship types, each one a form of dedicated weapon platform. If these changes can't be attached to a ship type, or if doing so would take too much work from making one for each race's unique ship style (and one of each for each ship size), then maybe just adding the more powerful weapons to each tier of weapons development would do.
For Mass Drivers, you would have the Spinal Mount type ship. Mass Drivers placed on it would have a huge mass spike, since the weapon is now going the whole length of the ship, but in return you get a correspondingly large increase in weapon damage, range, and accuracy, though probably with a significant decrease in fire rate. The point behind a spinal mount is that the whole length of it is a magnetic accelerator, so anything shot through it would go immensely faster than the turret hardpoints in the normal ship designer.
The missile ship would have internal missile storage bays and missile batteries - increasing mass used to increase rate of fire. Since missile are already the strongest weapons in terms of pure damage, I don't think further balancing would be necessary. Of course the logic here is simple - more missiles launched per salvo via the missile batteries, higher ROF, but bigger storage bays to account for more missiles.
Energy weapons would have a ship with oversized power supplies. Again, large Mass increases in exchange for increased damage and rate of fire. The energy weapon capacitors would be charged faster and to a higher firing level, but range is unaffected since that's the focusing array's job - in other words, power supply has nothing to do with it, it's all to do with the weapon.
In the tech tree, this would require an extra, one-off technology near the bottom of each weapon line. Spinal Mounts, Missile Batteries, and Oversized Reactors. Or if the weapon changes can be attached to new ship types, then they would be in the tech line that lets you build ever bigger ships.
I was actually surprised I couldn't find a mod that did this already, but I suppose the job would require significant balancing to make sure each oversized weapon didn't break the game.