Hello Faith and I think Cloe with real world experience on the matter has it right. But as your daughter is 9 and probably going on 20. Give her the respect and talk to her about it, see what she really wants to do. Lay the arguments out if she wants them and does not just give you the hand like mine do. And later on when she changes her mind respect her then too. It is more about being a parent then any title you can give your self.
Is you ex being unreasonable? Maybe but not really. She is only human to and wants to be special and herself just like you do. So let her be mom, that does not mean you are any less of a parent.