Richie,
How do you know that any of the tunes in Bunting relate to our ballad here? I can find nothing in Bunting itself that relates to our ballad or any of the tunes I know to it.

Bunting itself is quite difficult to navigate and it appears to be several volumes in one. On p46 in the main section there is a tune called 'Molly my Treasure' which would fit the metre of our ballad.

Also there is an awful lot of sophisticated art music in Bunting.

Then on p30 of a later section confusingly called Volume 1, there is 'Malli ban (Fair Molly) a very simple tune that doubled up could fit.

In the early 19th century Samuel Lover also wrote a very well-known song called 'Molly Bawn' 'Oh Molly Bawn, why leave me pining'.

In other words the title 'Molly Bawn' is practically useless unless you have other factors to go by in relating it to our ballad.

I still think your connections with Moore's poem are stretching things somewhat and there is a danger of wishful thinking. I'd need to see more evidence. Who actually connects the tune 'Lough Sheelin' with our 'Molly Bawn'?