I was thinking about something similar to invincibility frames. Shooting a gun should remove the invincibility "buff", as they clearly are ready to fight.
I should maybe add that it is less something for competitive, but more for pub matches. Spawncamping is incredibly easy on more maps than just Duel at Dawn, pub and comp. As rare as it may seem to happen for some, it's one of those things where players don't have fun at all and have no power to fight back, because they get shot before they can even think about shooting a gun.