I live in Atlanta, and my kids were all charter school educated and all went to boarding school for 9th grade. My eldest went to Tallulah Falls (class of 2014) and it was nice and close and a good fit (and very affordable). I know you said your family travels, unless something changes, they are sent home about one weekend a month for a long weekend. This may be an issue. It might have changed in the last 4 years as well, as they are undergoing incredible growth. They go to non-demonational chapel every week.
My youngest 2 went/go to Miller School of Albemarle in Charlottesville, VA. One graduated in May and we left the baby off there on Monday for the first time. We have been thrilled with what they have to offer us. It is a bit further (about 8 hours from here).
Let me know if I can help at all, i’ve done this a few times.