Laki, I store my hay out of our room, I hate it all over the floor. As long as you put it in a tote (and they make totes big enough for a full bale) you could store it in a snowdrift if you wanted. lol I would give some wiggle on where to store the hay, if the rabbit could stay in, especially since he's allergic. It's no fun being stuffy all the time.
Mike doesn't like my rabbits too much, they stomp at 3 AM, they dig all night, and they throw their hay all over the floor. Plus, the bucks smell. (Which I'm allergic to). He also doesn't like my African Clawed Frogs. But, he finds them useful when he gets an overabundance of guppies, and I stamped my feet and threw a tiz till he let me keep them here (I had them before I got kicked out)
We compromise a lot. The frogs are in a 5.5 right now (that I have to clean 3x a week) just so I can keep them in the limited amount of space.