Synge has a character sing lines from it AND ends the scene with it (stage direction).

Belfast, you're not being pedantic, as that implies you are also correct. In this case you might not be.

Synge went to some lengths to research his "west coast" plays - the dialect and speech patterns etc. I don't say he always got it right, but it would have been relatively easy for him to ensure that any song he put into a character's mouth was one that the character might have had some chance of knowing.

(I wasn't mentioning the reference as a factor in dating the song of course, just as a matter of interest. It obviously pre-dates Synge by quite a while.)