My thing is OP has an issue with the state of the in laws house, how MIL will happily stay up till 3am drinking, something about them having a cat. Oh and MIL has had the audacity to buy a cot when OP said no.
Is anyone thinking unless conceived alone, Dad has a say too. She hasn't said how she's feeding, just that she wants her own bed. I'm suspecting waking up in the messy house isn't her idea of a relaxing Sunday. She hasn't mentioned pain or any issues, just issues she has with MIL for wanting to spend time with DC.
There will be times when you cherish your MIL and the offer of a bit of help, I think someone said she's doing the childcare for you. It'd be a bit off if the contact was a convenience based subject. Oh you can have DC whilst I work, but no other times.
I think we often forget Dad's have a say too. Your options were he goes alone and you stay in with cat and baby, you both go & then you get a taxi home, or you stay the night and discover life isn't all that bad after all. Unless you'll be breathtesting all who want to hold baby, who may be slightly hungover.
What happens when OH is let off his leash? Does he get up as normal first thing to do baby stuff or do you cope ok solo & he takes over later so you can have a bit of a rest?
It does sound like your issue is MIL and your boundaries. But I fear seeing as it's been mentioned she's being utilised for childcare saving you a fortune, at some point you will have to compromise. It won't be as bad as you think.
The best way of putting it is, if it was your family, how would that play out?