edvardas, unfortunately, that wouldn't work; the Mormons liked Lincoln pretty well, and never really thought about rising up against him. Buchanan, though... If he messes up the Utah War even more, perhaps.
Maybe if you could keep the Utah War going for longer, into Lincoln's administration, it would spark more seccessionism.
But what's really needed is for the Mormons to see themselves as even more apart from the Gentiles; perhaps delay the Mexican-American war for a presidential administration or two. 8 years would give the Mormons time to have a more sizable population in the Utah region (in this case being a lot larger than Utah itself) So have the United States be distracted by something that isn't Manifest Destiny for a while, and let the Mormons build up their influence and power, and it might work. I think the maximum amount of time you could distract the Americans would be 3 presidential terms/12 years, and in that time, the Mormons could be a lot more numerous and able to resist limited US invasion. They could go the Swiss route, and use the mountains to make it not worth it to invade Deseret.
Also, get the native tribes on the side of the Mormons, and you could make the Utah War bloodier.