I never thought of a HS gym class. What a great idea. Do you partially homeschool your son in addition to his attending the small private school? Sounds really ideal. I think that's the best of both but haven't found a similar situation here.
I homeschool DS6. He goes to the private school only for 6 hours per week. It's enough to give us a break from each other and allow him to interact with the same group of children (mixed ages) on regular basis. I don't really rely on their curriculum. All I am asking for is plenty of social interaction. We found the school via our local hs newsletter.
I too am an introvert and fortunately so are my children. You should see us after a playdate. We all retreat to different rooms in our house, happy to be ALONE. LOL. Don't get me wrong we all like playdates and need them but we also need a quiet time after them.
Our hs group not only offers gym time but there were also ski trips, regular ice skating sessions, and hikes. That said we have yet to try any of the above. Gym and ice skating is on the next year schedule. You may also ask in your local gym and see if they have programs for homeschoolers during the school hours.