Wes, I like where you're going with this. Somewhat akin to the old Mid South league.
In Texas, they split private schools. Most play in the Texas Association of Private & Parochial Schools (TAPPS). They have their own governing body, officials, you name it.
Some of the older and more academically-prestigious schools (ex. St. Marks, ESD, FW Country Day) are members of the Southwest Preparatory Conference.
I could see Tennessee following one or both of these paths.