To do this, the Big 12 should first invite 7 schools, that would play for the right for the final two spots in the "Big 12." Thus, the term "Big 12" would take on a new legendary meaning.
Those first 7 invites would be: BYU, Boise St., Cincinnati, UConn, Houston, UCF, and USF. That first year, they play a 6-game "conference" schedule. The top-2 get in to fill out the Big 12. If there are ties, the contenders would meet in a bowl game(s).
The following year, the Big 12 adds Memphis, Temple, Navy, Air Force, San Diego St., Colorado St., and UNLV and brings along the "American" conference name. These 7 schools join with the remaining schools that didn't get the Big 12 call-up and separate into 2 "American" divisions, with an 8-game conference schedule - plus at least 1 "out-of-conference" game against a "Big 12" opponent.
Following the season, the two Big 12 bottom dwellers play the top 2 "American" division winners in bowl games with promotion/relegation on the line.
Olympic sports are simply divided by region. Navy as a football only member makes divisions a little tricky, but still workable.
SOUTH: Texas, Texas Tech, Houston, Baylor, TCU
NORTH: Kansas, KSU, Iowa St., Oklahoma, OSU, Memphis
EAST: WVU, UConn, Temple, UCF, USF, Cincinnati
WEST: BYU, Boise St., UNLV, SDSU, CSU, Air Force
Schedule to play those in your division H-H and then rotate a few games with other divisions - accounting for only 5 schools in the South division.