Oh Yuck to weak GT programs in public schools. I am convinced some of those programs are in place to appease parents and look good. We are in public and GT program in later elementary is wonder. My DD6 is in GT Math and it is a joke.
I am leery anytime a school tells me the can adequately accommodate my kids in a regular room. It depends on what the means. Do they group the classes by ability or just attempt to differentiate in the class. I am not a fan of the idea you can easily differentiate in the classroom. So really amazing teachers can do a good job, but in all reality most will struggle to adequately do it for the GT kids. Ask the school specifically how they accomplish the task. They are willing to keep the Math in place for her maybe they are willing to do more.
Given it is a private school, I am not sure they will do testing for you or that you would want them to do it. I doubt they do a lot of it and I would be concerned about an inexperienced tester with an introverted young girl. Consider looking for a GT experienced tester. I allowed our school to test my DD, but I asked the psych to spend a considerable amount of time with her first. She gets very anxious with new people and needs to establish a comfort level with a tester. The tester was very experienced working both at the school and in private practice. Ask them if they will take a look at outside testing. ACT has benchmark guidelines you could share with them. Your DD is at or above the benchmarks for an 8th grader on track for college success. They must be familiar with ACT even if not explore. It's a continuum from one test to the next.
http://www.act.org/solutions/college-career-readiness/college-readiness-benchmarks/
