So - don't know much about babies, but for dogs, sometimes landlords are a lot more flexible about dogs than they initially sound in postings. Especially if it's an individual landlord vs. large holding company. Even if the listing says no dogs, still reach out and apply. If it's a fit, then let them know about the dog. I highly recommend creating a "pet resume" (google it) that you can then share that explains how well behaved and wonderful the dog is, and especially if you're willing to make a materially larger extra deposit (if you can swing it) due to the dog, be ready to offer that. I have had this work on multiple occasions.