Ambrose Bierce; Devil's Dictionary: MUGWUMP, n.: In politics one afflicted with self-respect and addicted to the vice of independence. A term of contempt.

wofa, my uninformed guess is that what you mention is, as you speculate, an urban myth. I'm not familiar with the "vicar of bray" usage, and bartleby gives the etymology as "Massachusett mugguomp, mummugguomp, war leader".