While I knew the texas folk song version (completely reduced to the core, as many texas folk songs are) from a book, I heard a wonderful irish(!) song that might be the origin... the melody seems totally different for the moment, far more splendid and with some strange harmonies, the lyrics have twisted and changed a lot, but - blast I don't know the correct english technical term, in germany it's "aufbau" - the structure how the elements of the song are arranged is the same. And the short refrain seems simply to have been eliminated by a musician with bad memory.
It starts with "a frog went out to woe a mouse..." and the nonsense-phrase seem to origin in the stanza "far o'er the hills and bogs rode he" which repeats in all verses allthough it doesn't fit with the verses' texts. the irish tradition of singing "whack-a de liddy" and things like that seem to have killed this stanza by and by.