Are you sure that if you send him to private K they will actually put him in 1st next year? Is it just their word that you have or is there something written that says that they will do that?
It's such a hard choice because even if you send him this year he'll still be way ahead of his peers. My son (5) (as I'm sure a lot of kids here) could have started Kindergarten two or three years before he did and still would have been bored. I can't say that his level of boredom would be any different if I'd have had to start him a year later, if that makes sense.
The school said that they will work with you to give him a teacher that will be good for him? Have they said anything about differentiation? Will they move him up for some subjects? If they do will they only move him up one grade or will they consider moving him up to his level?
