Researching Beamish Family in the area of Drimoleague and Drinagh

The short version: I'm looking for information regarding my great-great-great-grandfather Richard Beamish, probably born in or near Drimoleague and Drinagh parish circa 1838. In particular, if possible, I'd like to identify his father.

The longer version: here's what I know, and how I know it. As I said, my great-great-great-grandfather was Richard Beamish, born in Ireland, probably Cork, probably in or near the parish of Drimoleague & Drinagh (I'll get to why), in roughly 1838.

Richard's mother was Eliza Beamish née Williamson, born c. 1810, daughter of a William Williamson. His father I don't know anything about at all, aside from the fact that his surname was presumably Beamish. If we go by Richard's eldest son's name, then Richard's father's name may have been Frank (or Richard), but that's a wild guess.

When he was young, Richard's father, whose name I don't know, died, after which his mother Eliza married again, this time to a John Nagle. This 1847 marriage record, from the parish of Drimoleague and Drinagh, I do have, and it's attached; it notably refers to Eliza Beamish as a "Protestant widow". They had a son, William Nagle, born in 1848 in Drimoleague and Drinagh, whose birth is also attached (and a few more Nagle children as well, but this one is the easiest to track).

John Nagle, his wife Eliza Nagle, and their children, both Nagle and Beamish, then moved to the United States sometime between 1850 and 1853. They are recorded in the 1855 census in Kings County, New York (attached). From this point, I can keep reasonably good track of the entire family through all the various US and NY censuses; I don't particularly need help tracking them in the United States, except insofar as there could be some information that would help me glean something about their origins in Ireland.

(The only slight difficulty I had tracking them was finding that Richard Beamish went by Richard Nagle when he was young, before he married Eliza Beamish, no relation [well, hopefully], after which he started going by Richard Beamish again. For some reasons I'll go through and some I won't go through here, I'm quite sure I have this right, even though it's a bit confusing.)

John Nagle, Eliza's second husband, died in Brooklyn on August 2, 1864; his death was announced in the Brooklyn Daily Eagle that day (attached), and he was referred to as "a native of the Parish of Drienaugh, County Cork, Ireland". I've found a NYC death record as well.

Richard Beamish, my great-great-great-grandfather, died on April 17, 1882; his death was also announced in the Eagle (attached), and conveniently the announcement notes that the funeral is to be held at the house of his brother, William Nagle. Weirdly, the NYC death record that lines up perfectly with the announcement in the Eagle has quite a few inconsistencies in it with what I know of Richard, but since I know I have Richard Beamish's address right, and since I know he was buried in the same plot as his wife and children in Green-Wood Cemetery, I know I have the right man, and the NYC death record could just have errors.

Eliza Nagle died on November 23, 1898, also in Brooklyn (also attached). It's only based on this death record that I believe her maiden name was Williamson, and that her father's name was William Williamson.

So! Based on all of that -- what information is there that could help me identify Richard's father, or learn more about his mother Eliza Williamson's origins? What kinds of records are there in this area (assuming it is Drimoleague & Drinagh, or failing that close by) that could help identify Richard's father? Would his death be recorded, or is this prior to the onset of death records? Any help would be appreciated!