My D16 fences. BUT, she was much like the OP's son... She tried several sports over the years, but that self-conscious, self-critical component was a real handicap. She did not start fencing until this last year. I wish we had started a year or two sooner, but really think if we had started any earlier than that, her self-defeating attitude would have killed it anyway.
We just kept a consistent message that she has to do something athletic, and stick with it for a full season once she signs up. I do think that sports that can be played more for fun worked best. For several years she took tennis lessons with a teacher who taught all levels of kids and did a ton of fun drills with them for most of every lesson. It wasn't a winning/losing/match thing at all. That was a good fit, until she got old enough to play on a team. Then she didn't like it.
She does love fencing now. It is cerebral, as mentioned above, and many of the kids who do it at her club are pretty geeky (like she is)
