I'm going to buck the trend here.
When I read the beginning of your post I assumed he was working full-time and thought that he had a point.
I too have a young baby, waking a few times a night, and it's hard, I totally agree but if you don't work on your relationship now it will only get worse.
However, he is being a total lazy arse not working but expecting you to do everything.
But, I don't think he is being awful in feeling this way, at all.
I think he has approached it wrong, but he is feeling ignored, and understandably so, you don't even share a bed.
Things need sorting, but BOTH of you need to work on it.
I cannot recommend sleeping sidecar enough, DS2 wakes up at 10:30ish when I go to bed, has a feed, then I have a cuddle with DP, who comes up at the same time as me. DS then wakes again about 1:30 and at about 3:30/4ish, at which times I can just pull him over to me, feed him lying down and practically sleep through, then when he is back to sleep I just push him back onto his cot.
What time does the baby go to bed?
Could you try ordering take-away tonight, and having dinner together so you can talk rather than argue?
Explain to him you're absolutely shattered, and that you understand he is feeling neglected but you're feeling like he is taking the piss and as he is at home you think he should do more around he house. Tell him everything that bothers you, but not in a digging way, don't say 'You're a lazy arse and you need to do more' phrase it with more emphasis on the 'I', so 'I am shattered, and I don't feel appreciated, so I don't feel like I want to have sex with you/sleep in the same bed but I can see where you're coming from'.
If he turns it into a conversation slagging you off, firmly but calmly redirect it. 'No, I won't sit here to be slagged off, I want to talk about how things can change, not be accused of doing things wrong, that way we can try and sort something out, but I am not wiling to let you speak to me like this'. Chances are he's not a complete arse, or why would you be with him anyway?
Also, I really think you need to start sharing a bed again, however you do it, because sharing a bed is about a lot more than sex.
(BTW, glad you go the BFing back on track, well done)