I think it is great that your son gets along with adults so well and is mature enough to be able to! I also think it is great that you recognize that there is a problem with him getting along with his peers!
I think you perfectly explained the problem there... he gets anxious around them. If I was in your shoes I would find a sport for him that allows him to be around more kids. I do not know if I would throw him right into a team sport though, but there is BMX racing(bicycle motocross- not motorized dirt bikes).
I had my son race from 10-14. It helped him to keep in shape physically plus helped to gain friendships with kids more his age and geared him up for better relationships with his peers without the anxiety he first had leaving our sides.... he was a major Momma and Grandmas boy!!! He is now a very independent 19 year old and is even thinking about getting back into racing. It really helped to build his confidence in himself and get over some of his original physical limitations he had! He also learned to control his emotions from racing and be a great sport when he lost... we learned the hard way to not let him win everytime we would play things with him!
If you go to www.ababmx.com you can find a track that would work for you. We had him go to the one in Elgin and Rockford as well as in the winter we would go up to Elkhorn, WI. If you wanted to bring your son out to the track in Elgin you can do that- or used to be able to- on any day that was not a race day to give it a try to see if he would even like it- bring a helmet and long sleeve shirt and pants... another thing we learned the hard way! He got on someone elses bike with bearclaw pedals and slipped and ripped his knee apart when he fell! Oops! I consider it a safe sport even though he had quite a few falls... nothing ever broke and the one incident we just taped it up since it was on his knee and nowhere a scar would cause problems later! He ended up with more injuries playing soccer and football later on in school and with broken bones and sprains there!
You might also want to check with your park district to see what programs they offer during the summer that might interest him... I am sure there are plenty!
Good luck and blessings to you!