How old is your DP?
You clearly have a problem with alcohol, and your child will suffer for it. If you are still up drinking now, you will still be drunk when your child wakes in a few hours time. That’s not safe.
Your DP sounds very controlling. But I also have to ask - why do you want to marry a man who regularly works 20 hour days? How certain are you that he’s actually working? If he’s working 20 hour days 5 days a week, he’s on less than NMW. You would both be financially better off for him changing jobs to something with better hours and you going back to work. Things are not going to get better - you will end up resenting doing all of the parenting, him never being home, no time for you as a couple. He must literally be walking in the door, falling asleep, then getting up and going back to work a few hours later. When does he eat? How is he functioning on so little sleep? How much travel time does he have, or have you included that in the 20 hours?
If you marry him, you are going to find yourself very, very lonely. Your drinking will get worse.
You mentioned a step child. How old are they, and is your partner home when they stop with you? Is he actually parenting that child or expecting you to do it? If leaving his child with you, how would that child’s mother feel if they knew you were absolutely wasted and in sole charge of their child? Flip it around to your child if needs be - how would you feel if you left your baby with a babysitter and came home to find they’d drunk a bottle of wine and half a bottle of string spirits? You absolutely should not be drinking to that extent if you are in sole charge of any child, but especially not someone else’s.
You need professional help to stop your drinking. Your partner has made it clear he will not support this. You should ask yourself why. Why would any partner be happy to see their DP become an alcoholic? Especially as he doesn’t drink himself and this is new behaviour for you. It’s either that he doesn’t actually know, or believe, just how much you are drinking; OR it makes it easier for him to control you.
It may also be worth counselling to help you with your PND and to get to the bottom of the cause of your drinking. I imagine you’ll find it’s that you are unhappy with the relationship. I know you said he wants a pre-nup (which seems odd unless he’s rich, which he doesn’t sound to be which is making kd wonder if this post is complete bollocks to be honest) and you weren’t worried as they aren’t worth the paper they’re printed on in the UK, but I’d be cautious still as if he has assets he may well be able to ring fence them (eg the house) that you wouldn’t be able to benefit from in a divorce. I’m guessing he’s been married before and it ended badly if he wants a pre nup this time?
Have you run a Clare’s Law request on him to make sure there’s nothing about abuse that comes up?