Please tell me you are on birth control and it is utterly bullet proof?
This is really not normal. He is controlling and abusive and is being aided and abetted by his mother. You have a 4 month old, not a 4 year old.
My husband cooked us dinner every night when my children were that age. Still does to be fair, he's a gem and a good cook. If he was home from work early enough he might do bath time, hand over the baby to me for a feed and go and cook dinner. I can't remember how we managed the cleaning but it was certainly a shared load and I probably did all the laundry as my colicky baby was keen to spew everywhere regularly so feeding the machine was a daily occurrence.
A daily group is not a lot, essentially this is you leaving the house for a walk or activity for 2 hours a day. Naturally with a 4 month old you've had to do this indoors all winter. So keep it up, maternity leave will be over soon enough and if you are sensible you will not be having any more children with this man. Fresh air is good for the baby end of discussion.
I suggest that he takes the baby and the dog out and he can listen to them both howl. You can use the time to "clean" since he's so desperate to have you swilling bleach fumes around with a nearly newborn. I wonder did SuperMIL have her own mother close by helping her that she has conveniently forgotten.
Your mum sounds like her heart is in the right place and she has your back. But her way of going about it is not helpful if she is going to put your partners back up and cause endless rows. He will just continue to try and alienate you from her and since she lives in a different country, she can bugger off home leaving you to deal with the fallout. I would tell her that he is not being kind to you and she is making it worse. You need to work through what is going to happen next and winding him up is going to make things worse so ask her to please stop provoking a row. I'd suggest you visit her for a bit over the summer and get some perspective from a distance.
You should confide in your HV so you have some real life support that will hopefully be constructive.
Longer term, ideally shorter term I think you need to consider your options re this relationship and get advice about leaving your partner. You don't sound as though you feel safe enough to tell him to FF off, his mother would likely support him with a 50:50 style custody relationship poisonous toad that she is. Ideally you'd move far far away from him but life is not that simple.
And enjoy your night out.