I think it's probably better to have this recognized by someone outside the family. If a health professional suggests getting out of the school, I think it's good to heed that advice.
I would be wary about just thinking the private school will solve all problems, especially at middle school age. We toured an expensive private in our area after hearing so many wonderful things from other parents, but were completely underwhelmed when we saw it. DS even said he would not want to go there. What is wonderful for some families/kids just isn't always a great fit for kids who are really out there. There are a bunch of threads about what to ask a school you are touring, I would suggest googling those before touring the private. Make sure you have a good idea of the things on your list that are nonnegotiable. Does the school group kids of like ability? Can your DD move up to a different class if the class she is placed in isn't enough if X subject?
Another thought -- are there any homeschooling groups around you? In our area, there is a decent-sized group of people who homeschool their GT kids, and they offer classes. I'm not sure what size city you are in, but if it's big enough, there is possibly a decent homeschool group you could connect with.
Good luck!
