I think my son sounds similar to yours in some ways.
My son was adamant he did not want to be grade accelerated because he enjoyed playing sport with his age mates. He is passionate about sport both as a participant and spectator. I think a part of it is the maths/statistics aspect of it.
He does best with organized sports where everything is fair. Playground time can be hard as it is crowed and disorganized.
My son also had a difficult time with this for a few of years. He expected lunchtime soccer etc to be played with professional rules. Unfortunately all the interventions instigated made no difference, it was something he eventually grew/matured out of.
So how did we find a good fit? Although we are in Australia perhaps you have schools of a similar structure near you. We are in a small rural town. The closest large town is 20 minutes away with numerous private and public schools with varying structure, so there was no shortage of choice. However we were very fortunate that the local school is a central school, public K12. The size of the school is about the same size as the nearby private boarding schools, Approximately 400 students across all grades. I believe size does play a part in allowing fexibility. My son was able to be subject accelerated with very little disruption. Initially when he started school he was extended in the classroom e.g when he started K he was reading at grade 5 level so he was given work appropriate for that age. Once he reached grade 3 all KLA's (key learning areas - english, maths and science) are timetabled concurrently up to grade 8 so the kids are grouped according to ability allowing him to work a number of years ahead. The school was able to continue this concurrent timetabling for him through to grade 12. I know there would have been no way the larger schools could have accommodated this. It was the concurrent timetabling that allowed my son to stay in his age grade as he wished and continue to be challenged.
You say that being a good friend has not always been easy for your son. By that do you mean he has had difficulty making friends?
My son is introverted and shy and from my perspective I see him as being on his own a lot of the time, even though he has his "sporting mates" he only occassionally mixes with them outside of school. I have worried about this in the past and continue to do so at times, but I wonder if it is my take on things as he is happy in his own company, at home anyway. Even though I know my son is content at school academically and athletically I know he has never found an intellectual peer there, at least not amongst the students.
It is such a delicate balance, compromise. How much will he really fit in socially given his giftedness and how happy will he be if he is not challenged?
I hope you find the answers you are looking for.