Well I spoke to Adam McN today and he was pretty adamant (excuse the pun) that it is one and the same song, written by Harry Linn, just that someone changed the title from "Fattest" to "Stoutest" (who knows, maybe because "Stoutest" has the additional meaning of being stout-hearted? Just my own speculation there.....!)
He did say that he thought that copies of Harry Linn's songs were only recently made available in the British Library, so maybe that is where the original lies.