Red flags are flying when I read the gifted school's comment. I wouldn't want my DD in that school because of all the schools you would think they would understand asynchronous development.
Question: Does his present school not continue on past K or is it not an option for other reasons?
Okay, more questions: In our state they won't consider Kindergarten placement in private school and would require the child to repeat kindergarten but if the child completes 1st grade in private the public school has no problem allowing them into 2nd no matter their age. It seems like your state is opposite. So why would this other private force him to repeat K? Can they not administer an assessment for placement?
Your two choices you presented are hard ones. I would hate to have my DD redo a grade when she is already so advanced yet I shutter at the idea of her in such big classes but this has more to do with her personality. If your son is a very social kid it might be okay. I really feel for you ... that is a tough choice.
