Yes, it’s called reclassing.

I have a 4th grader (and several older kids) in a top area private. How you are describing your child could be any of mine at that age, and they are/were in the pretty social groups, so I don't think you are realizing that your son is likely exactly where he should be maturity wise. Perhaps the school you are leaving has a particularly fast group or lots of older siblings or something?

Talk to the admissions office before submitting your application. They sometimes will put you down for grade 5/6 (or whatever the two grades are.) They will also know the birthdays of current students and can tell you where your son will fall in the range.