Her school won't provide any additional testing because she is fine academically and isn't enough of a behavior problem. I can just as easily see her fitting onto the spectrum, or having some kind of sensory issue rather than any kind of giftedness. I think we are at the point of just wanting to know how best to approach her problems.
Gifted girls are
notorious for hiding their abilities in school, like my DD did. This can then lead to other behaviors escalating, like perfectionism and meltdowns (again, this is what my DD did as well). Giftedness often looks like spectrum issues, or goes right along with it. Sensory issues and perfectionism are common with gifties, too. So I'd say, the first step in understanding how best to approach her problems is to get her tested properly.
As for the expense, I've seen several posters describe how they found a better deal by contacting the psych department at their local university, because grad students need practice.