Try to see this from the perspective of the school: if they're going to advance someone, they need evidence showing that he knows the material he's skipped. Everyone has to take the test to prove they know the material. If the school starts making exceptions, people will perceive an unfair application of policy, and the school will likely find itself with requests for more exceptions.
I think there's more to school than just absorbing material. For example, learning how to deal with stress is an important skill, and taking tests is one way that kids build their stress tolerance. Maybe staying in the class he's in now will help him learn to take exams.
It also sounds like his anxiety and pace got the best of him ... I can’t take another year of crying and school avoidance because he is not feeling challenged. ... The other complicating factor is that he is in a private, competitive school and I worry that if we leave we won’t get in anywhere.
You may wish to think about your own motivations here. It sounds like your son isn't happy at this school and that the environment there may not be conducive to good social and academic adjustment. Is he attending that school because of your ideas about where he should go to college? If so, please consider that attending an elite/top-tier college isn't a prerequisite to success in life. You may wish to read
Where You Go is Not Who You'll Be.
My kids are a little bit older (13+), and one of the things I've learned is that, as helpful as this board can be, it also has a way of getting people wound up about stuff --- like acceleration. Acceleration can be a good thing, but IMO, its virtues are overpromoted here and its very real downside tends to be dismissed (often in an off-handed way, as, "
yes, of course accleration doesn't work for everyone, BUT IT SURE WAS AMAZING FOR ME/MY KID 
").
