Thank you everyone!
You have been a sounding board for my own inner concerns and brought to light some things that hadn't occurred to me.

Yes, we can enroll him next month and pull him before August if he hasn't improved in the next few months. I do think 5 months could do the trick. We are going to enroll him in some extra classes now, to help with some social stuff. Classes at our Children's Museum, etc. Not too much though.

I do think holding him back, then placing him in Kinder next year is a very bad idea. He would be very bored and just cause problems. If we did hold him back, we would try for 1st grade next year. Though I am not a fan of this idea.

We have found two private schools, one is for gifted kids, that we will use if public doesn't work. They are costly so we will start saving now, just in case. If not private, then homeschooling.

The kinder here is not play based. It's a full on classroom. I have observed it. That might work for DS. He could do worksheets all day! Plus they have art, music, and PE. He takes Taekwondo and will take piano. He's already a good piano player with no lessons.

I will be working with the school counselor. I know they don't do pull out, but maybe they could do a gifted social class. They have some for other issues, like bullying and decision making.