I'm not an expert, but to get a quick clean kill you will need to spend more than that. Other options might be a slingshot, wrist rocket , bow and arrow, maybe even a blowgun. If you know someone that has one maybe you could borrow a more powerful air rifle. You might even be able to get a live trap for around your price range.I just wouldnt trust a BB gun to put a rabbit down fast.
Kill da wabbit!