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It appears that up to 1911, your Cooneys lived closer to Castlebar rather than Claremorris. The first link has the 1905 civil marriage record for Martin Cooney and Honor Walsh. Martin's father was Owen and Honor's father was Anthony (deceased). Martin Cooney was living in Deerpark Lower townland near Belcarra in Drum civil parish.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1905/10156/5695307.pdf
I checked the 1911 census and found the likely record for Martin's father Owen in Deerpark Lower. I also found Martin and his family in the nearby townland of Knockboy.
http://www.census.nationalarchives.ie/pages/1911/Mayo/Breaghwy/Deerpark_Lower/716217/
http://www.census.nationalarchives.ie/pages/1911/Mayo/Clogher/Knockboy/711841/
I was unable to locate a death record for Owen but his wife Honor died in 1912 and she was a widow at that time so Owen had to pass between March 31, 1901 when the census was taken and when Honor died.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1912/05362/4494958.pdf
I looked for a death record for Martin and the best lead is this 1949 death record for a Martin Conney who died in Castlebar Hospital. No guarantee this is him but the age fits.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1949/04549/4199963.pdf
As far as where the Cooneys were buried, I assume it was near the RC church in Belcarra. Here is the link to the parish newsletter for Balla and Belcarra parish https://communityballa.wordpress.com/ I would contact the parish and see what info they can share. Possibly the Cooneys are buried in Elmhall cemetery..
Roger McDonnell