My suggestion would really be to find a good time to talk directly (face to face) with her mother.
Make sure you couch it nicely and tactfully - expect her to be really hurt and upset - DC's can be so harsh and ungrateful for all the things that Mum's do for them - esp the things that they don't like as they usually are the ones telling off/setting rules, etc. Sorry for generalisations.
Having a child together means continuing contact and a need to be civil, as you say. And, don't forget this is for years to come - think what happens when your DC have DCs - your ex will be there as a grandparent too, so this is a really long relationship you will have to work on.
Plus, your daughter's not even a teen yet and that could be tough too.
Anyway, yes, talk to the mother as grown ups. She may know the answer, if not, she'll hopefully appreciate you trying to make sure you parent together as adults in cahoots, as you would if you lived together. Never decide these things with the kids directly, that's not fair on the kids or the other parent.
Tackle it together, the right way by cooperating and presenting a united front.
Whatever is up, is really important to her, so don't ignore it, but there might be a way to tackle it without changing living arrangements.
Living arrangements might be the best way to change things, but if she's just trying to escape the other home, it would be best to fix the problem (if poss) first.
Sorry if this is long and rambling.