I think if it turns out that he ends up with the house, it woudl be because he was awarded it at trial, because you're not agreeing for him to get it without the custody issues worked out, right?

OK... assuming that's true, when you go to trial, make it clear that the house needs to sell so that you can take the equity and move to a school district that's cheaper for you to live in. If that is what the judge has when he decides who gets the house, THEN a move would NOT be a change in circumstances.

Here's a similar situation. My husband knew, when we were dating, that our relationship was likely to continue. We decided to make it clear at the time of the divorce that we were planning to get married. That way, the ex (who everyone knows has been pretty evil with the way she's messed with the kid's minds and used every trick in the book to mess with hsi custody just to hurt him) would NOT be able to claim a "change in circumstances" just because we might get married. She would not be able to claim that this was bad for the kids at the time we said "I do", if she knew it was on it's way and did not mention it at the tiem of trial.

That's what I'm suggesting that you do. Make it abundantly clear that you will have to move and the kids will have to move with you, that staying in the overpriced neighborhood if it means living with the trashy girlfriend and her juvenile delinquent kids, would be worse for them than moving with you. And make that clear UP FRONT, so when it happens it will NOT be a change of circumstances.