Privates MAY listen to parents if they need the tuition income, but we found privates rather rigid. They have their own vision and it's great if that fits your kid, but if not, oh, well. Privates though have great marketing to make you think it's a good fit, so be wary of what you are looking at.
So much of this depends on individual schools. We started in public school and eventually landed in a private school and it's been absolutely wonderful - a school where teachers listen to parents, where differentiation occurs both naturally in the curriculum and specifically. Our school is also very up front when they feel a family/student isn't going to be a good fit, because it's not good in terms of future pr and advertising to have a family pay tuition, send their child to the school, and then go away unhappy about their experience. On the flip side, we have two other private schools in our area that are extremely inflexible about curriculum etc.
As mentioned above, you're not likely to find any school that's a great fit for a PG student. OTOH, we've found that while curriculum and pace may not be ideal, being able to be with peers who are at least MG for the most part made a huge difference for our kids, so if you can't find an ideal curriculum/differentiation fit, consider classmates - and in our area (which may be quirky), you don't necessarily have to go to a "gifted" school to find "accidentally-gifted" classrooms (in terms of peers).
I'm also curious why you would have to move? For a private school? Is it distance commuting? If so, could you give the school a try with a long commute first before going to the time/expense/etc of moving, just to make sure it's all that you are hoping it will be.
Best wishes,
polarbear
