Our eldest was on the bubble as well, late August birthday, we were dealing with a public school system though, on the other hand, at least we knew the school system exceedingly well.
Our eldest was exceedingly mature. Our biggest concerns was that by red-shirting him it would drive him nuts (as it already did) being around even less mature students AND that he'd be differentiated for. We decided to red-shirt him as we could pretty well assure he'd be reasonably differentiated for and if not we could supplement it at home. The issue of being around other kids of less maturity continued to bother him, however, subject acceleration at least gave him a chance to be around older kids a couple of classes a day. In the end I think we made the right decision. He was somewhat of the peanut of the class had he not red-shirted.
In your situation, if your son's maturity level and stubborness seem to be the issues....and if you're confident that even with red-shirting him he'll be properly differentiated for, then I'd encourage you to do so. I fail to see the draw back. Another year at home will give him time to mature, time for the family to address the issues at hand for a year at home, and it's never fun going through school as the "runt" of your class. There are enough issues raising a GT child without fighting the physical factor too IMO.
