Like was said above, both have conferences for all the other sports outside of football. Texas would need the same kind of situation. BYU doesn't mind having their non-football sports in a smaller conference of religious-based universities, so that works for them but Texas has too much pride in their basketball, baseball, volleyball, golf, tennis, swimming, etc., to be ok with putting them in Conference USA or the Southland Conference, or anything like that.
They'd likely have to find a power-5 conference that would agree to take ONLY their non-football sports the way the ACC did with Notre Dame. And truth be told, the "football deal" of having to play said conference 4 to 6 times a season (Notre Dame has to play the ACC 5 times) wouldn't be the world thing in the world... the problem is, which conference would agree to that? The Big 10 never agreed to that kind of deal with Notre Dame. It's doubtful they ever would change their mind on that. I doubt the SEC would agree to that kind of thing either. The ACC seems the most likely to make the deal (since they did that with Notre Dame already anyway).
Unfortunately there's a problem with that. The administration at UT has been pretty vocal about the issues with sending our non-football sports halfway across the country on a weekly basis. Football is one thing... it's almost always Saturday, and there are only 12 to 15 games a season. Every other sport has week-day games regularly and has a lot more games or meets or matches or whatever. They miss class. They miss even more class when the event is in New Jersey or Massachusetts or Washington. And if you're just adding the non-football sports to the Pac 12 or Big 10 or ACC, you're likely not helping make arrangements to bring "travel partners" with you. If you're joining full time to one of those conferences, you're probably looking at bringing other Texas, Oklahoma, or Kansas schools with you, which helps make the travel at least a little bit better.
It's just a really complicated situation, and unlike Notre Dame, we aren't as close, distance-wise, to some of the members in the less stable conferences. I know there are TV issues that would need to be worked through, but when it's all said and done I think we're most likely to be in the East division of the new Pac 16, or in either the west division (or some kind of pod system) or the Big 10++ (which might be 18 members large or something like that). The SEC and the ACC seem less likely, but the SEC might be the 3rd most likely, if for no other reason than the ACC just makes it that much tougher to expand in a way that makes sense travel-wise if the administrators are at all serious about looking out for non-football travel. It gets really weird trying to come up with a way to add Texas "travel partners" to the ACC, while one could imagine 2 or 3 out of Kansas or Oklahoma or Rice or Tulane joining the Big 10 with Texas, along with looking into Notre Dame.
Here is a quick easy fix.
Texas football goes independent, and leaves the rest of their sports in the Big 12. Not sure how the big 12 would like that? It's sorta like being married to a girl who just flat out tells you she wants to be married but also wants to sleep with who she wants. Most schools would probably say no to it.