As much as I don't like the association with public school, the Homebound Teacher seems like it fits your situation best. He would be enrolled at public, he would have a schedule of classes, and the HT would bring the assignments to your house, and pick up the completed ones, and maybe sit with him while he takes the same tests the rest of the class takes. If he goes back, he's already enrolled, already has classwork done and tests taken, no muss, no fuss. If he doesn't go back, but he's enrolled for the year, maybe he could do that academy thing next year. He would be responsible to someone besides you -- he would be responsible to his classroom teacher, through the HT. And yet you can be there to answer questions for him whenever he has some and the HT is not there.
That is what I did when I worked with the school district. Depending on how many classes I was tutoring determined how many days I came out. For a couple of kids I was there 5 days a week and handled all their classes.
