I doubt assault rifles are acceptable technology for the Amish too.
I wouldn’t put it past the Mormon army to use nukes, but I doubt they need to, and the population density of Amish isn’t going to give us the megadeaths we want from nukes anyway
My money’s still on the Mormons in a conventional fight, especially as the Amish (as you said) are opposed to fighting anyway
The Mormons even have a nascent arms industry. A bunch of up and coming gun and silencer companies come out of Utah and Idaho.
Finally, the Mormon church has literally 100 billion dollars, impassable mountains with simple chokepoints, trained veterans with combat experience, and zero issues with using the latest technologies.
I would go for Mormons. It looks like they have a high amount of troops which are relatively close by. This means that you can mobilize big parts of your Army and simply Overrun the first Libes of the Amish defended quite quickly. The Amish troops are quite spread out so help would probably come to late.
Not entirely sure who would win, but from this map I can see they both clearly hate ocean water/coasts. Mormons seem to be slightly more accepting of it so maybe that gives them an advantage. Adaptation is key to success!