Sunday play has been a major issue regarding an invite for BYU to the Big 12 (this coming directly from the mouth of a major TV guy with knowledge of the situation).
(1) the Big 12 does not want to invite BYU for "football only".
(2) quite a few Big 12 ADs are against appeasing BYU and stopping Sunday play for non-football, non-basketball sports (i.e., baseball, track and field, etc).