Your mil doesn't respect you and your dh is an idiot. No wonder you don't want to let her have your baby over night if she won't even communicate with you when you have asked her to babysit. It's weird.
I don't have my baby yet, but my mil is excited to be a Nana soon but she's very lovely and would never ask or demand something like this. Which makes me more likely to feel comfy about asking them if we need it. Not that i would... I don't know if they'd want to overnight anyway. It's kinda moot, we love too far away for that to happen unless we are staying over too!
Anyway. I would say to your mil that if you need an overnight babysitter then of course she can have the job. But right now you don't need that and nor does your baby.
Maybe sit down with just your mil and just explain you feel that everything has got out of hand and you appreciate that she really wants to help but you are fine at the moment. All her to stop pressuring your dh because it is hard for him.
Although she doesn't sound very reasonable about things so not sure if that'd with.
Alternatively sit her down alone and say that after all this stress and demands that she is the last person you would EVER ask to look after your child EVER again. And that unless she shuts up about it that she will have no contact.
Then ask your dh if he finds it hard living with a baby so much that he's willing to send her away, that maybe he should stay with his mother. And give you some peace and quiet!
Is be quite mad i think if my oh suggested such a thing, unless it was something we both wanted.