The tune called the Lonesome Boatman has no connection. Whether the song or the usual air of Fear a' Bhàta is ever known as The Lonesome Boatman I don't know. There is no reason that it should be, as it is the lover left ashore who is lonesome in the song.

As for being a version of the Irish song, Dónal Óg; Fear a' Bháta is included in a book about the many versions of Dónal Óg because of a verse in common, but I would not consider them versions of the same song. There are admittedly a couple of other things they have in common; in both songs a woman laments that she has been abandoned by a lover named Domhnall (Dónal is a modernised Irish spelling).

Another inaccuracy in Amy's 2009 post is from John Loesberg's Irish Ballad Book, which was plainly in need of a good editor and a decent researcher (Amy herself could probably have done a better job researching and Amy said in 2009 that she was 17 years young). In Scottish Gaelic bàta means a boat and bata means a stick; the à indicates drawing out the sound longer. In Irish Gaelic bad means a boat and bata means a stick, but the Rathlin island dialect used in the song collected in County Antrim has a heavy Scottish influence, including the word bàta for a boat. And it is generally conceded that the song Fear a' Bhàta came to Ireland from Scotland and not the other way around (though to complicate matters, maybe Síne or whoever composed the song borrowed a verse from an Irish song!).

synchronicity: just as I type this I hear Caitlín Maude on RTE Radio 1, singing Dónal Óg. If she had lived long enough, Caitlín would have been 80 yesterday (22 May).

btw, Fear a' Bhàta and Fhir a' Bhàta are both okay. If you are addressing someone, saying to him, "hey Boatman" you would say "[a] Fhir a' Bhàta. But to say The Boat Man, the Gaelic is Fear a' Bhàta.