what my rights will be with the children.
Not sure I understand what you mean by this, but the bottom line is that by going there, you are dependant on his say so when it comes to moving back - cutting both ways, and no you can't get some sort of agreement that if things were to go wrong you could come back with the children. Well, you could, with him - but if he chose to disregard it there's not much you could do. Bottom line - once they're resident in the US, and they would be - in school, etc. - you could NOT bring the m back to the UK without his consent.
So -
It goes brilliantly for him, he loves the job, you hate it and are miserable. You say you've had enough - he refuses. You're stuck.
It goes wrong for him, he hates it - you make friends and love it - it's his job you're out there for so he can unilaterally decide you're all coming home whether you want to or not.
He is working - meeting new people, caught up in a whole new life. You are at home with babies. He works long hours. Lots of pressure, as you want support from him as you are now doing it without your circle of friends and family. He feels pressure to put in the hours and prove himself in new job. Things are strained betwen you, he meets someone else and something happens. Ok. So whereas in the UK you could tell him to fuck off - here, you are stuck. You can't leave with the children, he won't give you permission to go. You don't have enough money to move out. You probably couldn't even move out and separate as visa requirements might mean that you have to be together - a particularly horrific move would be him informing authorities you've separated and your visa being revoked. He has the power.
Basically, this kind of move when you have small children creates a very, very unequal balance of power. It might be ok, but if it were ever to develop into a situation where it wasn't ok, you'd be fucked and he'd hold all the cards. That in itself is an issue, by gonig, you put your welfare in his hands.
I would never put myself in that position.