Hi Jon,
John has just sent me a copy of his sheet music which is just a song that mentions them singing 'Rosin the Beau'. It has the same title and is set in NY but is obviously based on an earlier song. My gut feeling says all of these 1838 onwards pieces are very likely based on an earlier RTB, be it British (Irish) or American.

The other piece to follow. The 1838 again is obviously a new version of an older song.