Finding the right placement in school is hard. When my DD was your child's age, she was the same way. We never had her IQ tested because I said "what would it tell me, besides a number?" I knew she was crazy smart, and I went with the approach of working with the school to accomodate her needs. Though we had no formal IQ testing from a private psych, she has scored 99% on every standardized test she's ever taken. She has taken out of grade level testing and still scored just as high. She has been grade skipped and guess what.... she's still not challenged. She has a 4.3 GPA for both quarters in the skipped grade. She isn't challenged, but she is happy, and isn't bored like she was before this skip. While I can't say it solved anything, it did move us ahead to a "somewhat better fit". I can guess we'll be in need of another skip or some accomodation at some point in the future. Right now DD is happy, and that is what I base my need for a change on. She complains that it's easy, but not like she used to. Take it one step at a time with your child. Remember that you don't need to do all the moving and solve all up front. Take it step by step. I look at these moves as stepping stones along the path of where you need to follow her. One skip, see how it goes, when more is needed, advocate for more. Don't look for the solve the whole problem with one answer approach. I am certain DD would be on the same path as yours if we kept her moving at the pace she was capable, but, she is happy and gaining other things at school besides just academics (which she really isn't getting from school.) There isn't a day that goes by that I don't question when we need to do something more again. I also don't agree with the, don't teach her anything approach as your psych suggested. I never taught DD anything just to teach it, it was usually lead by her and I followed. You can't stop their desire to learn, just put on your running shoes and getting ready to run to keep pace with her
Good luck!
