Sooooo... you'll risk being *that* classmate, but yes, you can. If a faculty or staff person asks you to leave the animal outside, you can say that it is a service animal. If they ask what for, you can say mental/emotional health. You'd need to attest that it is a legitimate service animal. BUT, the faculty/staff person cannot legally ask you to produce proof that it's a service animal. It'd be blatantly obvious that you're a liar, but this is a route you *could* go.

If there is a classmate with an allergy, then I believe which one of you has to leave would be at the discretion of the professor.

Your best bet, though, would probably be to find a friend to chill with the puppy for a while.