BYU joining the PAC-12 has been a discussion topic for 10+ years. But it will NEVER happen, especially in today's climate of intolerance to views contrary to one's own.
PAC-12 reps hide behind the reason that BYU is not a "research university", only an undergraduate school. But that's politically correct cover...the real reason is its religious foundation, not playing on Sundays, taking positions against social issues that conflict with California's secular administrators, etc. PAC-12 will never have a religious institution in its conference, not even if Notre Dame came calling. (btw, that eliminates Santa Clara, Pepperdine, LMU, USD, Gonzaga, to name just a few others)
Conservatives can't even give a speech on Berkeley's campus without a military-esque group of security and a protest of thousands because students feel threatened by someone advocating a pro-life position. Can you imagine the outrage when the LDS football team would come to town? (99% of students at BYU are Mormon.) Folks like Ann Coulter talk a lot of nonsense (and a lot I disagree with btw), but it' s not hate speech. And she deserves the right to speak anywhere/anytime. Just as AOC or others on the far left deserve the same. It seems that we should be able to separate sport from politics.
Big12 would be a better fit politically, especially with other Christian schools already in the conference (TCU, Baylor) but an interesting nugget is that Big12 has NO men's soccer in the conference. Although football and basketball drive the decisions, men's soccer would be eliminated.