The original 'Barney Brallaghan's Courtship' was a London stage song written in the mid 1820s by Thomas Hudson to a tune written a few years earlier by Jonathan Blewitt, sometime musical director of the Theatre Royal, Dublin. It's still occasionally called 'Blewitt's Jig', though the great popularity of 'Barney Brallaghan' and its sequels (there were several) ensured a change of name in most cases.

There was a thread some time ago which you may like to look at: Lyr Req: barney brannigan. That was actually a completely different song, mind. The link I provided there to more info is broken just now. You can see broadside of Brallaghan and his relatives at Bodleian Library Broadside Ballads. Use the 'browse' function to locate them; stsrt with 'barney' and then try 'judy' [Callaghan].

There is sheet music of 1830 and another undated edition at The Lester Levy Sheet Music Collection: just search for 'Barney Brallaghan'.

Of course, I have no idea what the Doonans may have sung.