The coach may have no idea that this kid is taking things so personally. I try very hard to talk to a girl about
her gymnastics, but there are always those who are so hard on themselves that they assume their coach must be disappointed in them and they interpret any correction as a personal attack.
I don't know the coach, but from experience, the most frustrating situation to deal with is an athlete who saves up hurt feelings and resentments for a day when she can't make a correction and then unloads it all. I don't know that it's in any way conscious...but it does happen.
It's funny that you think the coach thinks that your daughter is okay with not moving up. Why would you/your daughter think that, if she's not happy?
Maybe this frustration is coming out during workout without saying anything, and the coach is reacting to the body language and lack of effort.
It sounds like the bottom line is that your daughter needs to accept the fact that she is repeating a level and try to benefit from it or move on.
Switching gyms may help, or it may be a quick fix...and it sounds like you are not the kind of mom who would want your daughter to just run from a tough situation.
I would never recommend doing privates at some other gym to "try it out". If you are committed to your current coaches, then show them some respect and talk it out before going to try a new gym. Then leave instead of playing games.