Really bluntly - you have a lovely life because he is a high earner, so you can enjoy these years. But his life right now is anything but lovely. It sounds a bit shit. And if you were him, and your life partner said, no, stay in the job that makes you incredibly unhappy, because it bankrolls my life - how would you feel?
I absolutely agree with the previous comment that you and he need to establish that he'd be less stressed, and more fulfilled, in the new role (as far as such things can be known). But if that seems probable, then something has to change. If you don't move, he needs a career path that doesn't cause him such misery, and that may mean less money and a less comfortable lifestyle.
I think you also need to establish whether, as implied, he'd have a better work life balance at the new job. Who says, on what evidence? After all, if you all move over there and just never see him, because he's even busier, then that's an issue and a half.
I also think a few years in Australia or New Zealand could be wonderful. Think of the travel you could do, the outdoors lifestyle such young kids could enjoy, and the quality of life all of you could share. That's not rose-tinted specs; I'm a dual national myself. I wouldn't want to live there forever, but I would love the chance of a few years out there while the children are relatively young. Investigate it. You could rent your house out for the duration and enjoy a family adventure, and then see where you are.
I would think carefully about what your choices here could mean. If he's that unhappy, it's down to being in a job he finds frustrating and past his level but which takes every waking moment he has, and you refuse to move to a place he can work at something more challenging and interesting, then it's not his career you aren't supporting. It's his mental health.
One final proviso, though. Australia is very strong on not allowing people to remove children from the jurisdiction, if there is a split. If you decided you weren't happy, or he met someone else, and you wanted to return to the UK... if your customary domicile is Australia, then he could stop you. It's unfortunately not unknown. If your marriage is shaky, then sadly you may want to investigate that aspect carefully. People crying that you should just go for it should reflect on whether that's a prospect they would welcome. I don't know if you can legally agree between you that their customary domicile should remain the UK, and that it is in their interests to be brought up over here, and have such an agreement hold any water in that situation. But in your shoes, I'd want to look into it.