Sorry you're having such a hard time OP, DPs family do sound demanding but it could be that they're just very keen to be fully part of the baby's life.
Could you explain to them that you're still not feeling very well after the birth and would find it tiring and stressful to stay overnight anywhere except your DM's at the moment?
I think if you make it about you rather than the baby it may go down better. Most of us forget just how disorientated we were in the early weeks of looking after a first baby but can sympathise if someone has had a tough birth.
You could emphasise how much you're looking forward to staying, once you're up to it, and leave battles over the dog, atmosphere, and where you'd sleep till you're feeling stronger and have agreed things with DP.
It'll make life much easier if you can both agree on how you'll handle both families. Perhaps you could find out what your DM is doing to get his back up and tackle that in return for him supporting you about the overnights?
When a new baby comes into the family it can throw up all sorts of conflict - just when you're least equipped to deal with it.
I know it's difficult when you want to feel protected by everyone at a vulnerable time but I think it's very rare that that happens in real life.
Relatives on both sides do feel they have a stake in your child and that can be so hard to cope with. The alternative though, is having family who don't really care and have nothing to do with your DC - I know which I'd prefer.
If you can maintain the boundaries you need to look after yourself and the baby but bite your lip and avoid saying anything you'll regret until things are calmer, you'll be better off in the long run.
Good luck,