Question: Would you not provide money if she moved? Even though she'll be in a different country, you could still send money to her online (I would get it clarified she'd be ok with this first - sending it unprompted would suck). That way you could help provide financially while still putting their needs first.

Of course, I don't know you, and so anything I say will be built on presumptions. But if you cheated twice, that sounds like poor impulse control, and perhaps a lack of emotional maturity. I feel like your daughter should live with the parent who'd best raise her, and it sounds to be her mother - by your own admission, she has a stronger bond with her mother.