A boxing match, with a moat running through the middle of the ring. If both sides are pragmatic, then what you would see would be raiding parties on isolated areas. Most deaths would be from the privation and hardship of trekking across hundreds of miles of desert and mountain. If one side isn't pragmatic, then the result will be an army perishing somewhere in what is now the Navajo Nation. It is hard to emphasize just how incommunicable the Wasatch front is with the settled Mexican areas, even the ones on the fringes like the New Mexican settlements and Tucson. Remember that around this time period, it took Mormon pioneers months just to hack a barely passable road to the Blandings/Monticello area in Southern Utah. That's still several hundred miles from the nearest Mexican outpost.
One caveat: if the Saints make a real claim for California and have a real presence there, fighting in California is a possibility.