Next time he sayss "I'm off out for the day on Sunday, you can't stop me, it's my choice", you say "that's fine. I'm off out for the day on Sunday too, that's my choice. Who have you got lined up to look after the children?"
Pictish, the difference between the OP and her DP, is that he tells her he is going out and assumes she will be looking after his children for him. While she wonders if she will be able to go out, and thinks she will ask him to babysit his own children, but half-expects him to say no. His behaviour is only ok if it is ok for her to behave like that, too.
OP, I really think you should go this Sunday. Not because it will be fun, but to show him that he has to consider other people, and dads have responsibilities too, and what your world is like. You won't have to struggle with the pram, because he can do that, can't he? And the kids needn't get bored, because you can send them in the same room as their dad and give them brushes. The Great Decorator can pass on his skills
. And then you can always go for a stroll with the baby. Just the baby. And ask him for some money before you pop out - he's not going to say No in front of his family, that would make him look selfish ...