I believe, from what you say, that you should maybe think about therapy for the two of you before you agree to this...

It seems to me that your financial decisions (not you personally, but how things are split) are not quite right and you have some resentment for that, in addition to how the house chores are split.
You also should maybe figure out if he's doing this simply to prove a point to his ex-wife...I think that needs to be brought to the forefront, too.

How will you make her pay her portion? Will you both agree on what is to be done about consequences?

This is, indeed, difficult. I can see both his point and your point as far as having his daughter live with you. And, no offense, but you do see how this marriage is affecting you, right?