Good Luck!
Have you had a chance to observe the kids in the classroom at work? Lots of private schools say that their kids are working at one year plus grade level but I'd want to see it before I believed it.
If the school wants your son without the skip, see if they will let him start Mid Feb or March for 5th as a trial, so that if things don't go well, they can make him do 5th again next year (or you can go back to the public school with your skip in place.)
DS15 did public 4th, then private 5th for a month, and had his first official skip into 6th in October. I don't remember if we needed teacher recommendations, but they wouldn't have been strong back then!
Do you have any other private options? Keep looking!
Grinity