While Seán Tyrrell's tune is much sung and presumably much liked, it is definitely not the one which Frazier had in mind (and I think most artisan poets of 18th 19th century had a tune in mind as they wrote). However, at the time, in all parts of Ireland there was much use of Scottish tunes.
Try the tune for "A Man's a Man for a' that." It brings it within the area of songs which can be sung unaccompanied.