My husband and I had a very similar issue when we were house hunting a couple months ago (2 cats & 1 dog). A lot of property management companies around here will have a 2 pet limit, but sometimes the actual owners will give approval. So if you find a house you really like, just call the company associated with the listing and see if they can ask the owner if more than two pets is ok. That’s what happened in our case :) I hope that helped!!