1. BYU and the church have never come straight out and said that playing sports on Sunday is breaking the Sabbath. I believe their opinion is "decide for yourself". They keep themselves aloof from the controversy by simply stating that they will not play sports on Sunday, thus allowing those that have chosen not to, to not have to.

2. If they agree to a game time, they would be contractually liable to play the game start to finish. I imagine they could throw a tantrum and protest a game that went past midnight, but then the question, "why did you agree to such a late start time?" would come up. If anyone thinks BYU would win that fight in the court of public opinion, they're fooling themselves.

3. Why do we worry about such things? BYU has a whole athletic department, and they play late games all the time. Does every single person that asks this question think they're the first person that has ever asked it?