You may wish to check the policies of each school. Many gifted schools may have a student body which is in the moderately gifted (MG) range as this profile is simply more common. There are fewer highly gifted (HG) students. There are even fewer profoundly gifted (PG) students.

The school's admission criteria and policies may reveal whether the school may tend to have intellectual peers for your daughter.

I am creating a link to your thread on a similar topic: District gifted program vs. private school