Your son and my son sound like the EXACT same child! My son is now in first grade w/ poor fine motor, SID, impulsive behavior, and being silly. He loves to get everyone to laugh!! =-)
Nathan didn't have private kind., so you can definitely do what's best for your son, but here's what we did...
He is the one of the oldest in the class (fall bday) so he wasn't technically "held back"... in public kindergarten he wasn't challenged at all, and he was not in the "top" groups b/c they didn't recognize his intelligence (this will NOT happen with your son!).
In first grade, he was immediately promoted to the top reading group and is with peers who are reading a grade and a half ahead of level! His math group is advanced too, but his math is not nearly the level of his reading.
So for Nathan, he is put in groups to keep his intellect stimulated, but socially he is still a little "silly" and "young." We had to adapt his homework b/c in first grade they require the child to write FIFTY words in one night, and that was painfully slow... now he only has to write 30 and I write in highlighter and he traces it.
For your son, kindergaten could be way too easy, but on the flip side, 1st grade is a HUGE jump... they are writing paragraphs, sentences, doing spelling tests, fractions, etc. They don't have centers anymore, or free play. First grade was very demanding on Nathan had a horrible time transitioning. I made a big stink about the JUMP from K to 1, but all the parents said the same thing... TO help him adjust, they gave Nathan sensory breaks.
Your son is older (summer baby) and has good experience w/ private K... so he could be a different story. Our son has benefited from the extra year of social skills, and has not suffered academically. But go with your gut, and maybe give yourself permission for a "trial" if you want to try first grade...
BEST OF LUCK!