Jesus, the people who say "leave him" are always very trigger happy on Mumsnet.
Clearly this isn't a situation that can continue. Your DH isn't going to divorce you because you insist he takes on more chores and contribute fairly to finances. If he does, you shouldn't be with him.
It's fine to maintain two separate current accounts. But you have two children now, you need joint accounts for joint expenses and joint savings. This is an absolute red line. It's perfectly fine for you to contribute 2/3 to outgoings while he contributes a 1/3. That reflects your income and that's fair. But you need a joint bills account and a joint savings account. None of this "he transfers me x amount per month". Both your names should be on the accounts. If you had this already, you could have drawn from your joint savings to top up your maternity pay.
Your joint outgoings account should cover rent/mortgage, bills, holidays and kids expenses. Personally I'd set up two accounts for this. One solely for rent or mortgage, as that's generally a fixed amount per month, and the other for bills/expenses which can fluctuate. I haven't read all the thread but I am assuming you are not the sole owner of wherever you live.
Chores and mental load are a bugbear for every married couple. However, seeing how you've set yourselves up financially, I can imagine this is probably a bigger problem for you both than most. Download a family chores/mental load checklist - there's loads on the internet if you google them. Work180 has an especially helpful one. Print out one for you and one for him. Often the problem is that couples can't see what the other person is doing - especially the hidden mental load of things like doctors appointments.
You need to take time out to discuss all of this with him. Do you have family or a friend close by who might be able to take your kids for a few hours on a weekend? Go out to a cafe and get breakfast with him. Don't do an evening conversation when you're tired and when potentially alcohol can get involved. Make sure the conversation happens somewhere out of the house so that he can't make an excuse that something else needs to be done. Practice what you're going to say. I suggest telling him this evening that you are both going out for breakfast in the morning as you need to sort out finances and X has agreed to babysit. Don't get into a conversation then and there - tell him that it needs to be discussed without distractions. Hold your line if he tries - "I'm tired, I don't want an argument, I just want to watch tv. I am serious when I say I am not talking about it now".
When you get to the cafe, tell him how you want the conversation to go. i.e. "we're both aware that the split of things like finances and chores have been a source of arguments for the both of us. What I want is that when we go home today we have split finances and chores fairly. We don't have to both be 100% happy, but we need to have an agreement. I have some suggestions, and I'm very open to whatever you might suggest too. But I have to clear, continuing the way we are is not one of the options".
Don't have any patience with the nonsense of "I do more than any guy at work". You didn't marry his workmates. And unless he steps into a time machine and emerges at his workplace in the 1950's, he's bllshtting you anyway. Just keep bringing the conversation back to the both of you. You can shoot the breeze on how you were raised or who he hangs out with AFTER you have come to your agreement. Oh, and by the way, before you leave - and assuming you haven't poured your coffee over his head - say "this was so helpful. I feel so much better. It's so good to get away from the kids for an hour or two and actually have an honest conversation. I'm definitely going to ask X if she can babysit again in a couple of months and we can see if this is all working." That puts him on notice that there's going to be follow up.