I think the easiest way to get an "American Taiping Rebellion" is to incorporate your "God Worshipping Society" allegory, bet it the Mormons or some other religious group, into the already existing political divisions of the Antebellum United States. For instance, maybe, for whatever reason, Mormonism becomes popular amongst Southern elites, and when an alternate American Civil War breaks out over slavery, there's a distinctly religious element to it where the Southern states' leadership are disproportionately Mormon and view the conflict as a war both to maintain their slave-based socioeconomic system and as a religious crusade to spread their beliefs to the rest of the country. Alternatively, you could have Mormonism just become a lot more popular out west to the point where an uprising out West is on a much larger scale, but that's realistically never going to reach the scale of a full-blown takeover of the US or conflict on the scale of the Taiping Rebellion.