Zach, I hunt a lot, and often bring home rabbits. Several years ago I contacted a fur tanner to get information on how I could turn these skins into something. I was told very simply that rabbit skin is too thin, prone to tearing, and that fur loss was a strong risk as well. I abandoned the idea at that time.
I do see your interest, in that there are many fur sporrans on the market that are made with rabbit (real and synthetic) fur.
Anyway, I guess this is not too helpful. Good luck with whatever you are thinking about trying.