So I want to do a photoshoot using airsoft gear and using a forest as a background, would it be possible here? Who can I ask for permission?

Plenty of room and forest around but like Chris said call and ask

Depending on what part of rhe Forest you want to do the shot at you also have Pyramid Lake. But call or talk to your local Fish and Game Warden this is Federal lands so most likely you will need a special permit for photo shoot

Just call office if permission is required.

Sure.just call, to the office.