Really I don't see the problem with having lasers be faster than light. Yeah, it's unrealistic given the Theory of Relativity, but nothing else in the game follows Relativistic physics. It's not that hard to get missiles, or even ships up to a significant fraction of c, yet my ships and missiles don't get heavier at higher speeds. I can communicate in real time with aliens who aren't even in the same system as me. Enemy beam weapons don't hurt more if I am moving towards them.
Missiles are capped at lightspeed, too(no, the design wasn't practical. Also a few versions back). And presumably so would ships.
The rest actually makes sense for an inertialess drive(and would explain why kinetic kill missiles aren't a thing and ramming does so little damage), which Trans-Newtonian engines seem to be, in terms of function and the idea of trans-newtonian mechanics(with matching real velocities handwaved; presumably there's an inertial drive mode or something with much lower performance, or they just loiter in the gravity well or something). Well, except for FTL communications, but that's something rather different than FTL beam weapons. FTL beam weapons would probably be their own technology, possibly an exotic one you have to discover before you can research it, assuming super-long-range beam weapons were desirable from a game design perspective.