You're the mom! If you're evaluating all the evidence and you think a grade skip is what's right for him, then I think you do that. No question.
I always think about reversibility and risk--there seems to be fairly little risk with the skip. If it isn't going well, you just pull him out and homeschool, as you would do if you didn't try the skip, right? No real risk there. And if he wants to go back to school or to the independent school next year, you could send him later, with or without the skip after a year out for homeschooling.
Follow your gut. We're talking in generalities; you're talking about your specific child. YMMV!
