It'd be kinda awesome and a kinda awesome status to have "beat entire game with no offensive attack." Add a bonus of like 1,000,000 Drew Carey points. (You know, where everything's made up and the points don't matter.)
You need it in level 3 though because you have to throw stuff at switches to flip them on and off. I like that idea though. I could add a score bonus for completing level 2 though without picking up a weapon.