This isn’t about the dog. You are brand new parents and instead of focusing on making a loving family, you are fighting about pets. That suggests to me that neither one of you was really ready to parent a child. Yet, here you are with a month old baby who needs the two of you to be in a loving, mature relationship.
I’m assuming you both decided to give away the cat and keep the dog. That your husband is making a big deal about the dog now (after the baby was born) looks to me like he’s fighting for control. Perhaps being a father scares him. Maybe he doesn’t like sharing you. You’re not likely to give the baby away, but maybe on some level he thinks that making you give your 4-legged child away will guarantee that you still have time for him.
You are responding by taking him literally and dealing with his objections to the dog — as if anything will please him enough. It won’t work because the fight isn’t really about the dog to begin with. It’s about growing up and taking responsibility for decisions that the two of you made and figuring out how to be parents as well as lovers.
To solve the impasse, the two of you need to quit talking about dogs and start talking about how you are going to deal with all the absolutely reasonable fears and dreams that go with committing to have and raise a child. If you can’t have that discussion on your own, I suggest you find a couples therapist — quick — to help you out.
I wish you well.