I am an immigrant to the US, albeit not from the UK. The main issue I see is that you would be financially vulnerable because of any delays in ability to accept a job in the USA because of your visa status. Once you are married, you cannot apply for a green card (via marriage) for at least a couple of years, so you'd have to work out what visa status would allow you to work, or else you'd have to go without employment until you got a temporary work permit when you applied for your green card. (If you decide to move, I would suggest you get an immigration lawyer to tell you your options there so that you can minimise how long you have to be unemployed)
I think you would also be legally vulnerable because of having established a domicile for your children in the USA. I don't know the details but again you might want to talk to an immigration lawyer in the USA before you make a decision so you know what your situation will be.
So in theory you could end up with no job, and if your marriage doesn't work out, no green card, and no legal ability to take your kids back to UK to live.
Your fiance on the other hand, would not be risking his job. Would be getting to see his kids, and would be establishing the US as his children's official place of residence.
In other words, YOU would be taking on the majority of the risk and your fiance would be taking on much less risk.
He would however be taking on a small amount of risk after 2 years because once you apply for a green card through marriage, he would be agreeing to support you financially for at least 10 years. Again, please confirm this with a US immigration lawyer as it was years ago I dealt with this. But to get to that point, your marriage would have to survive the few years until you could apply for a green card, and he would have to actually follow through with the green card application.
Having a green card also means you'd have to live in the USA 6 months of the year to retain it, or, if you switched to US citizenship, you'd be liable for taxation as a US citizen.
Why can't your fiance move to the UK instead?