That's a big leap. Imagine instead that Auburn was the one and only university for the Southern Baptist Church. Imagine that it's primary mission was to educate members of the church, and to evangelize and convert others. That football players stopped playing to take a "mission". That's what BYU is. We're not talking about a Catholic college, or SMU being Methodist, we are talking about a school that is a true extension of the LDS church.
Now consider that they already have a sweet TV deal with ESPN as an independent and their own TV network. Why do they change the code that reflects their core beliefs again? I don't think they are going to budge. They don't need the Big 12.