You mileage may (will) vary, but we found a grade skip at the same time as a move from private to public school worked well. Our daughter was going to change her classmates with either, so doing both was not any more social stress.
The public school system seemed amiable to the idea that a gifted, private school taught student might be ahead of her age mates and were happy to evaluate her for a grade skip.
Set up a tour of the public school and while you are meeting with an administrator, bring up the idea of acceleration.
Best of luck,
--S.F.
