I haven't even been able to trace the melody back as far as Heilner and '22, just the lyrics.

afaik the oldest music is the 1927 Atkinson arrangement. Same songbook and Shaw-McCutcheon family tree as John B.

fwiw the above, Heilner, Le Gallienne, Prouty et al are all one step removed from the Bahamian Tourist Board and nothing in that songbook is "original."

Still looking...