Do NOT enable this person by quitting work. Just do NOT, even if it means splitting. He does not see parenting as anything to do with him at all, real parenting, not Disney daddy-ing.
Any occasion where he is by himself with the kids (hasn't happened yet since DD2 arrived), he sees as me getting one up on him. This is where the reluctance to help comes from, as far as I can tell.
Help? Help? He sees his own children as a tit-for-tat, point-scoring exercise with his own wife. Parenting your children is not 'helping'.
He seeks to punish you, full of resentment that you're at home and he's toiling in the mines.
That will get worse if you pack in work to stay at home.
You can't change him. Best you can do is stay equal in terms of earning and work so he has no ability to skive out of lifework.
The comment about 'then I get a weekend', too, is telling. He resents having his own children imposed on him because they are your job.
The only way forward is to keep paid employment. Full time. And nail him down about who is doing what to do their share of looking after them and lifework, including holidays and time off for illness if you are not using a nanny.
It doesn't bode well, OP.
DH was once the SAHP. We were a team, equals who chose to have children. And we acted that way.