Okay, hmm. I think for avoiders and I used to have these tendencies it helps to put a proper schedule in place, it makes it less avoidable. So instead of saying, can we swap a few nights, work out what you think an equitable plan is, for example he does night feeds on Sunday, Wednesday, Friday. That way he can also prepare for them in his own way.
If he is really only working 4-5 hours a day then he can obviously take naps during the day to catch up on sleep.
Also, I think for avoiders, the fear is worse than the reality. So if he really refuses to do this, then I would second the idea of going away for a day or two and letting him get on with it.
Did he do night feeds with DS1? Is he remembering it as much worse than it was?
I don't think you should 'ask' him to do things, but present a plan for shared tasks, and if he disagrees with any of it, ask him for a counter-proposal, and try to get a compromise. I mean, you shouldn't have to do this, he's being really selfish, but do what you need to do for now.
Finally at the risk of being flamed, it is not the end of the world to put your DS in his own room if he is keeping you awake all night. I had to do this quite early for similar reasons, many people do.