I tried scrolling through the other suggestions and didn't find it, but was thinking how about a HQ attack that uses attack timer can also benefit the gun a bit? Maybe 0.1 or 0.05 but just it adds a bit, as you're learning to deal damage and knocking a whole building down so you should be able to get a bit of attack knowledge. That's all. 

HQ attacks doesn't use your gun. it uses either a molotov or dynamite.