A reader writes:
You wrote, "So far no readers have found a precedent for someone with that high a religious office running for president as the nominee of a major party." James Garfield was a minister for the Disciples of Christ, which I believe is a more significant role than "high priest" in the Mormon church.
More on his religious service here. The other major example from readers:
William Jennings Bryan (Democratic nominee 1896 and 1900) was an elder of the Presbyterian Church and an active preacher for much of his life. After his presidential runs, he sought to become Moderator of the General Assembly (the highest position in the Presbyterian Church), but he narrowly lost.
More background on Bryan here.