When I was on mat leave I usually managed to cook, but that was it. I told exP looking after the baby was a job I was doing all day and it was hard work. That included taking the baby out to groups, weekly HV clinic and just meeting friends, because that was good for the baby and for me.
ExP didn't do much housework so... it didn't get done and the place was a tip. But I absolutely don't think it's the job of someone who is looking after small children to also keep on top of housework at the same time. Ideally you would do equal shares at evenings/weekends and get it done that way.
Yes absolutely if you choose to be a SAHM when your DC are in school, it's fair that you should fill that spare time with contributing to the household by looking after the house. That's an entirely different situation from looking after a baby, ESPECIALLY at 4 weeks.
Him going out at the weekend for 8-hour stretches is totally not on. That's when he should be giving you a break and doing his share of chores. Or if he must do that, then you get the same amount of time off too.
There's one good cure for your husband and that's to leave him with the baby and go out for the same amount of time he does - a full work day plus commuting time. Give him a list of chores and tell him you expect the chores done, a tidy house and tea on the table when you get back.
Of course harder if your baby is BF and I wouldn't do it at such a young age, but this is how to make him realise what it's like.