It's hard give you and answer without more details about both schools. That said, when looking at the public schools, it's important to remember that they have no mandate or other formal directive to help kids who are ahead of the grade-level curriculum. Their entire reward and punishment system is based on the scores of low-performing students. I wouldn't expect them to be flexible in the way you've described, but they could be. Have you talked to them? What's their policy on grade skips, for example? What do class and homework assignments look like?
FWIW, I went to school in a college town. There were only two kids in my class of 50 or so whose parents were professors, and one of them arrived in 7th grade. Neither one of them was HG and probably not even MG.
