I don't know whether N. Carolina has this option, but our district in NM allows homeschooling parents to have their children come to school for specific classes such as PE, music, art, computer, etc. It gives the kids a chance to socialize with neighborhood kids in their school so that they're part of a school community without forcing them to participate in the full day of brick-and-mortar classes.
I second the recommendation that you get an IQ score; the more facts you have to support your requests, the more likely you'll be met with success.
