Sir, - The expression "deadman's chest" is probably a corruption of "deadman's face", oddly enough. The latter was a kind of swivel plate to which two anchors could be attached when a sailing ship was to be left at anchor for a long time.
Breaking out the two anchors took a lot of effort - hence the fifteen men". The term occurs in several sea songs and shanties, according to Stan Hugill, the foremost authority on shanties and a genuine old seadog himself.
As to whether RLS or an American published a version first, the most likely explanation is that there was a common root - a musical hall version of a shanty, perhaps? The dates tally, since shanties had outlived their true, use by the mid 19th century but were often metamorphosed into "minstrel songs", for example. The latter were certainly very popular on both sides of the Atlantic when these authors were alive. - Yours, etc.,