It sounds like you still have her in the parental role. She's paying for your wife's school, you're both depending on her for child care, and your wife is still desperate for her approval. It puts you in a position where you owe her. It's hard to set boundaries with someone that you are indebted to. I think you have two choices. Option 1: you become independent adults and take out a student loan for your wife and pay for daycare for your child. Millions of people do it. It's not as comfy, but it comes with autonomy. If MIL wants to see your child she has to do it on your terms. Option 2: you continue to listen to what she says, rehome your dog, and let her continue providing for you until your wife had finished her degree and can get a better job. There's really no middle ground with people like that.

I personally have a 6 month old and I wouldn't keep a pet that was aggressive. My daughter will sneak up in the cats, grab a handful of fur, and scream from excitement. All of our cats are super well socialized, so they just get up and go lay down somewhere else. They have cat trees, but they choose to hang out on the floor where the baby can get them because the they don't really care. We try to teach her to pet nice, but she's an infant so it's slow going. What would your dog do if that happened? Are you going to be unable to have other children over to play with your kid because your dog is unstable? I don't really know the extent of your dog's problems, so I can't give you advice. I can say that if you have doubts you should find the dog a new carefully chosen home and not just take them back to the shelter.