You're in a tough situation on two counts: your child and your health.
From what you've described, you've gone above and beyond the call of duty in trying to educate him. Sending him to a public school could be good for him. Kids need to know that there are acceptable and unacceptable behaviors, and sometimes they need an external source to teach them that lesson. As a parent, you can tell him stuff and advise him regarding behavior, but he won't have to learn lessons until his actions have consequences. That can't happen when you can't get out of bed. Teachers and other kids will provide the consequences in a way that a parent can't.
You have a right to be able to take care of your health, which means that you don't have to put your son's desires over your own needs. You'll be much better able to address your medical condition if you don't have the added stress of a defiant child getting in the way. He doesn't have a right to do that to you, and IMO, you shouldn't let him.
So I agree with your idea about sending him to public school. He's lost a privilege.
As for the gap in executive function and learning ability, I'd like to suggest something new.
Very poor EF may indicate that he isn't actually ready for high school-level mathematics. Being able to understand concepts and/or do the mechanics of the math is only one part of what's needed. Moving to abstract topics requires a different kind of thinking and an ability to be organized --- about how you think about the problem, how you structure your approach to it, and how you go through the steps of writing stuff down. It's essential that math be practiced daily, or the skills get rusty quickly. So even though he can do the work when he wants to, it's not enough. He needs to develop discipline for working
consistently. Plus, working only when you want to just isn't how life is. Sometimes you have to do the task NOW because you just have to, and that's all there is to it. He's at an age where some daily homework will help him learn that lesson. Even if the
work itself is easy, the part about
getting it done will be hard. So he can learn the second skill without sacrificing knowledge.
A lot of what we learn comes through facing adversity. Your post indicates that some adversity may be precisely what your son needs. I really mean that --- NEEDS.
Best of luck. I know it's not easy.
