I'd think the military response would mostly be for dealing with terror attacks, a reaction to the aliens slaughtering civilians and wiping out the local police resistance. For crashes, I'd suggest the chance/rate of alien-death be increased, and civilian death be reduced, since crash-aliens wouldn't usually be equipped or prepared to go on terror-rampages, and would already be injured/stunned from the impact. The military would probably arrive about as quickly, possibly faster having detected the dogfight and crash. Also, the "aliens-killed-without-PHALANX's-help" penalty should probably be smaller, considering PHALANX already did some of the work in shooting the ship down. Should the locals secure the site, though, they keep the craft and loot. Maybe if you piss that country off enough, they'll start using that gear against you.
As it is, the sniper rifle may be too strong; it may over-power the military units in-battle, especially in comparison to their poor performance on the geoscape. I think the following may create a system that would allow for the military to be perceived as using the more powerful end of military hardware, without actually having them easily overwhelming the aliens or, worse, out-performing PHALANX.
If civilian/local equipment can be easily excluded from combat spoils, I would actually suggest having a second set of equipment instead of the weapons available to PHALANX, but with lower damage or accuracy ratings. PHALANX gets the pick of the best equipment from participating nations, but the individual nations have their own, mass-produced, not-quite-the-best-there-is gear, especially when it comes to civilians and police. Thus the pistols, shotguns, rifles, etc., would be less effective than what PHALANX uses. The models of the weapons could just be recycled from the existing ones, so I don't think much work would be needed other than re-stating and making sure the locals draw from the right weapon sets.