My dc did ft hs with pt electives at a private school (where I also taught). It worked great for us, although I agree w pp that the "socialization" was not a plus & they were daily frustrated with how little time was spent on topic compared to how much time was spent dealing w student behavior & classroom management.
As far as I know, in the states, *ALL* children of compulsory age have a legal right to attend public school & to make use of the available funding & resources (I mean, it makes sense-- hs parents pay taxes that fund public schools, too, right?) It is hs that is legally restricted, not public school. So theoretically, it should be possible everywhere. Now, back to reality-- it really does depend on how cooperative the locals are; sorry.
I'd contact HSLDA-- they are awesome re answering HS legal questions. HTH.