Simple Answer: He doesn't hate/despise BYU. Seriously. As a BYU alum, he has affection for BYU, and an enormous amount
of respect and appreciation for LaVell Edwards.
BUT, as the HC of BYU's archrival, University of Utah Utes, he's got to walk a very fine line. His job, right now, is to make Utah the most successful they can be. He is not Kalani Sitake- Mr Nice Guy- "I hope everyone has fun.". Kyle succeeded Urban Meyer- one of the MOST intense college football coaches of all time. Urban referred to BYU as the "Team Down South". Some could very reasonably argue that Urban took it too far. (His intensity basically caused his brain to explode.) But nobody can argue with the success that he brought to Utah.
If Kyle had followed another coach- like McBride- maybe he'd be different. But he didn't. And part of Kyle's coaching style is to use everything he possibly can to establish a psychological edge vs. the other team. Doesn't matter if the opponent is USC, Michigan, Oregon, Alabama, or BYU. He wants his players to come out of the tunnel with a chip on their shoulder.
I kind of wonder how BYU's football players feel when Kalani wishes their archrival success- in the 11 other games.
When he retires, I expect his attitude towards BYU to soften dramatically. He's always going to be a Ute, but he will also show more reverence towards BYU.
The fact that he seriously considered becoming BYU's HC is all the evidence you need- to understand that he doesn't dislike BYU.