It is just a pun on “bat man”.

Adapt and come up with stuff on the fly? If by that you mean TPK when so much as a single dice roll diverges from my meticulously crafted grand plan, then I may be the greatest DM of all time.

Rabbits breed like crazy and multiply fast. Rabbits have been associated with fertility for centuries in pagan traditions. The rabbit is the symbol of the germanic pagan goddess of Spring and fertility, Eostre. I suspect there is a connection here.