It is unfair of him to not help with any of the childcare. He should be waking through the night on nights when he is off the next day - that would be a bare minimum. This then enables you to get a full night's sleep.
He should also be sharing the days off. If you both get two days off a week, you should each be allowed a lie in, taking it in turns. A proper lie in, too, not an hour of him loudly banging around huffing and puffing.
Fwiw, my DH works mon-fri and I am currently on Mat Leave. I do all the cooking and dishes, the lions share of the laundry and the lions share of the tidying. However, our house is not a show home looks sadly over at mountain of clean laundry piling higher than me
BUT when I get a lie in, I get up to a DH who has fed, entertained DD, put her down for a nap and is pottering on with laundry/ tidying or something else. He even took her out yesterday morning (my lie in day) as I was hungover and he thought I'd prefer a quiet empty house to sleep in. Today I did the same, and took DD to the park at 8am, got home at 11 to a very happy rested DH. However, he is popping to the shops today to buy a shirt, so has said he will take DD with him, meaning Ill get a little nap or a chance to run the hoover round. Whatever I choose, I know he won't come back and expect 'stuff done'.
OP, have you tried simply getting up early on one of your shared days off and saying 'goodbye' to DH and letting him get on with it? If you do, Id suggest picking a day you know your MIL is busy so he can't call her in to take over.