My problem with this WI is that rabbits eat gardens, breed "like rabbits", and taste good. All of this combines to make it an excellent idea to kill and eat rabbits.
I don't think this WI is an impossibility in the mid 1900s; as noted above, this is becoming true today.
Only European Rabbits can be domesticated, and not the American cottontails;
Not quite true. 'Tis possible to keep them in captivity, but they tend to die of heart-attacks, so they need very much space. This is not actual domestication, though. I've never heard of anyone trying to keep them through several generations.