"For clarification, the reason I pay all the childcare is that DH pays all the other bills." Except that he doesn't. Because "I can’t give up work as we can’t afford it" . So he doesn't earn enough to pay all the bills.
I think you are being played by this lazy fucker on several levels.
Let's start with the financial level. Housing - rented or bought? If rented, in which names, who is the tenant? If bought, are you on the deeds/mortgage? Because it's far too common for one partner to put all the house payments through their account, shafting the other partner by making it look as if they paid nothing towards it.
Is there a joint account? If both wages were going into one account to meet joint costs, there would be none of this I-pay-he-pays shite.
By making it that YOU pay childcare, I suspect it manipulates you into thinking there's no money left over for any of your personal needs (sanitary pads/tampons, new knickers, haircut, shoes,dental appointment etc.) and so you become super-frugal while he still gets what he wants.
And then there's the practical level that prompted this thread. He expects you to do EVERYTHING while he does as close to fuck all as is possible. "His exact words were ‘i need to relax at the weekend, and if I have to look after dd after a week of work it will kill me’." Well boo fucking hoo. Welcome to the real world of parenting. It clearly won't kill him, but giving in to his demands clearly will exhaust you. SO DON"T GIVE IN TO HIM.
There is one solution to this, and it is in his hands. He needs to grow the fuck up and become a supportive husband and father and if he can't manage that he needs to fuck off. He's not a single man with responsibility only to himself, he's a father with responsibility to his child and a husband with responsibility to his wife. He is NOT some sort of overlord to you, he is your team-mate and he needs to pull his weight.
Most times, I think the best thing is for couples to sit down and talk everything through, because usually the problem is that they've not discussed expectations beforehand and they've been talking at cross-purposes. On this occasion I thinkI'd be inclined to rip hip a new arsehole and tell him to act his age not his shoe size and to stop being a burden not a helpmate.