One interesting idea would be to have upgrades within a 'tier' of weapon technology.
First generation lasers are all bulky and heavy, as their power sources lack energy density- you get a heavy long-range weapon that blows a couple of holes in any earthly material.
When you reverse-engineer the alien plasma weapon power supply, you can miniaturize laser tech and get laser pistols, sniper rifles, and a repeating laser cannon that takes the role of the LMG. (No laser shotgun equivalent, since that doesn't make any gorram sense).
First-generation plasma weapons use the same power supply, but there's something about heat dissipation or charging time that makes them slow to fire. Mod 1 gives a weapon that spews out a bunch of mostly incoherent plasma in a mostly-aimable spread (filling in the shotgun role, but much slower than ballistic shotguns at first). With further research and refinement, the issues with keeping a plasma bolt coherent over longer distances in a man-portable package get solved, and we get the pistol, carbine, and HMG equivalents (leaving the laser sniper rifle as the best-in-role at that tech level).
Mag weapons combine the three: Accelerate a chunk of highly dense material at the target, keeping it in a single chunk using the technobabble of the plasma weapons, and heating it into a white-hot chunk of melt-their-torso-off as it leaves the barrel using a pulse of coherent light. Has the knockdown power of ballistic weapons while delivering the energy of a plasma bolt at the ranges of a laser, and adaptable into all popular form factors.
If there's enough types of tech to leave one of the major weapon types out of each level (except the first and last), it would add a level of stractical thinking and require that everyone be able to adjust to changing equipment availability, but in any case giving out an inferior version of one type early (and the normal version "on time") also rewards players being able to adapt to the equipment they have available.
Damage "types" only make gameplay sense if players can bring many different types into each mission in order to benefit from attacking where the enemy is weak. That would mean typically having multiple weapon types on the mission; if each soldier has only a main weapon and a sidearm, then the sidearm needs to be MORE effective in some situation to be worth using it for damage type diversity.