Go for it. I held my boys back from Kindy (same thing - small, late b’day) and that was a great decision but I can only imagine the benefits if we’d waited till a transition year.
We had friends who lived the next block over who moved to our town and held back their rising 5th grader - best decision they ever made, they said.
Here’s my only caveat - private schools have become incredibly hard to get into - where I live, since Covid, the wait lists are a mile long. Start working connections now and donating money NOW.