God save us from Student Politics! Madness!!
Ask them to furnish you a set of guidelines as to what defines a "sport" in their view. E-mail it to some of the people that have been suggested already, or post a link to it on the forum here, and I'll bet that in a few hours you'll have a pretty watertight definition of paintball as a sport. Plus it's going to give people here endless satisfaction in picking holes in their reasoning
Certainly if you push the tourney angle as Tony says, then it's ludicrous to think of it as a "society", but then you'll probably need some sort of established Uni league and regular match fixtures (but hey! what better way to inprove UK tourney teams than regular, subsidised practice and competition?)
Don't let the b******s grind you down.