By your description of your daughter that she is "strong-willed" and tends to talk back and disobey rules that SHE eels don't make sense, I presume that the difficulty here is that your daughter is getting in trouble at school? If that is the case, how would a change in schools affect this? You will have to show that the attitude problem that apparently exists from your daughter would almost certainly disappear at the new school. Otherwise, the presumption will be that the same problems can and probably will arise at the new school as they have at the current school.

It might also be possible that some of your daughter's issues with acting out are related to conflict between you and the ex. It is not uncommon for a child to act out in the way you have described and when mom and dad are at odds with one another. Is your daughter in counseling at all? If not, perhaps you and the ex should consider therapy. There might very well be some other cause for her behavior and the any behavior that might result in her being picked on. And, in my experience both in law enforcement working juvenile crimes and as an educator, it is rarely ever a one way street when a child is claiming to being picked on. So, I hope that you are not necessarily taking the position that your daughter is an innocent victim here - she may not be. But, there is likely much more going on, and the school she goes to may have little impact on whatever the underlying problem might be.

If you have the support of a therapist or some objective proof that a change in schools is very likely to be beneficial rather than a claim that it MIGHT be, that would help. But, a court may well want to see something more than what you have proposed so far if it will force an alteration in arrangements against the will of the other parent.

Good luck.