Something akin to a general manager of football who is much more involved than the AD. This guy is hired to run the program in terms of image, rules, off the field stuff, community involvement, honor code, and religious matters. Keep the "must be LDS" requirement for this position. This would be more than an AD--this guy would be in the locker room, in team meetings, and be a leader on the recruiting front. He'd have a football coaching background. Just to toss a name out there--a guy like Ken Niumatalolo (I said a guy like him, so don't just dismiss the idea because Ken specifically might not want the role).
Then allow he and the AD to go out and hire the best football coach out there, regardless of religion. Find a good fit in an now-expanded pool. Let the coach run the practices, make on-field rules, teach, scream, swear, and make all on-field calls, hire a staff of his choosing, etc.
I recognize that many coaches still wouldn't go for this because they wouldn't view it as "their program." Many men want 100% control and even prefer a weak AD that they can push around.
However, there are likely some coaches who wouldn't mind just being a coach and having someone else manage the CEO, off-field stuff at any school--let alone BYU.
Just trying to think outside the box. If BYU wants to compete, they can't begin with a realistic coaching pool of just 3-5 men. We got lucky with Bronco. BYU needs to cast a much wider net. I understand why BYU has the LDS requirement. Football is a very visible face of the school and BYU is a very unique place with a distinct culture and lot of extra rules. So let's figure out how to meet those demands while hiring the best possible coach.