While it's normal for kids to express wants or feelings, it's not a good idea to allow them to make decisions or requests when it comes to changing schedules.
I went through a very hideous divorce. We were in mediation often. It was made very clear to my ex and I that working out the schedules was between the adults. That's not kid territory.
For instance, if one of our son's friends sent home a birthday invitation that fell on dad's weekend, it was up to me to contact dad and let him know that our son really wanted to go, but, if they had other special plans already, our son would just have to understand. And, vice versa. My son was invited to things with kids near his dad's house and wanted to go, but it was also a weekend where we were having family from out of town. So, we could compromise if possible. If you have something planned for a month and something comes up with a week's notice, you can't always change things.
I know for a long time, my ex would ask my son what he wanted and of course my son would say what his dad wanted him to say for fear he'd be mad when in reality, what he wanted was to keep things the same because we had other plans. There is NOTHING wrong with being flexible. I've traded a weekend because there were historical sailing ships my ex wanted to take my son to tour. He traded weekends when we went to stay with my mom after an operation. We definitely got better at agreeing to things as we went along, but it was pretty much pounded into our heads that the kids aren't the ones that make the decisions. They aren't the ones in charge of abiding by or changing the schedules. That's adult stuff. Your daughter is only 9. It's not her job to make or change the schedule. Hopefully, you can talk to your ex and trade time so she can participate in something important to her. And if he wants to verify that's what she truly wants, that's when she talks to him. Not beforehand.
My son is 16 now and his dad and I still discuss changes first. We've been divorced basically my son's whole life so it took a lot of trial and error, for sure, but the children should never be the messenger.
My ex had a first wife and they let their kid do whatever he wanted to the extent he basically had no parental judgment involved because they both wanted him to like them best so anything HE said he wanted, he got. Neither one of them would dare tell him no to anything he wanted. He was calling the shots and he knew it. Being a kid, he didn't care if it was inconvenient or they fought about it as long as he got what he wanted. You do NOT want to set up a situation like that.
My ex and I communicated in writing as well as verbally. Everything was written down and unless it was some type of last minute emergency or unforseen thing, we had to give each other at least a week's notice, in writing. That was done to keep the child's schedule as predictible as possible.
Yes....sometimes things come up.
I personally do not think it's the child's job to negotiate schedules. I say that because of all my experience with the California courts frowning upon it.
My ex kept trying to say that my son only wanted to be with him because he's a boy and needs his dad. They used the example of some kids who will say they want to live with their father because he has a bigger screen TV or a swimming pool or they don't have to do homework while they're there.
Any kid worth their salt will want what they want and what they want isn't always best.
If your ex is agreeable to letting your daughter attend this party, then you have to tell him that you are willing to give up x,y, and z so that he doesn't M. any time with them. It's fair and equitable and something that you adults should work out. If he says no, he says no. I've told my ex no when he asked me the night before Thanksgiving if he could have our son when it was our scheduled time. That's why we went to giving each other advance notice and letting each other think about it without saying yes or no right away.
I mean no offense and I'm not saying you've done anything wrong, but your daughter is not the messenger or the decision maker and you'll be in for a world of hurt if you put her in that position.
Just my opinion and again, no offense, I promise.