Big 12 they should. But historically I don't know why that should be the case. Less than 1% of the state is LDS so not a huge draw there, and the majority of kids going to college probably don't want an honor code life.
Also, I'm not sure that most people in Florida give a hoot about BYU or its football tradition. I would suspect that given a choice between the two, most recruits from Florida would choose USU over BYU because of the baggage that BYU brings.