From what you've written, I disagree with the above posters and I think it is sounding like the school and AAP (GT) program are a good fit for your daughter, but she needs some extra support/modifications to succeed and thrive there. This is exactly why it is fully possible and not very unusual for a student to have an IEP while attending an AAP (GT) center.
This is just me, but I'd go straight to an outside evaluator, see what the results are, and if the results indicate that an IEP would be appropriate, then I'd start that process with the school system. With a child in AAP, you are going to have a harder time convincing a screening committee that testing is necessary toward an IEP, and I'd just skip that hassle and pay to have it done myself. But, if that isn't feasible for you, then start with the school.
I also agree with the posters who said that adhd should be considered here. It's worth a visit to your ped and maybe to a developmental ped to fully explore that possibility.
Good luck. I think you are pursuing some very worthwhile avenues to help your daughter thrive.
