In Breand Breathnach's Folk Music and Dances of Ireland, CÁILIN Ó CHOIS TSÚIRE MÉ is listed as the earliest notated Irish melody(1584). It is a variant of THE CROPPY BOY.

I don't know of any psalms, but there are a number of religious songs in the Irish tradition, many from the repetoire of Joe Heaney I will offer more discussion later when I have time.

I suspect you know a lot more about church history than I do. I believe that the singing of psalms was something that was started by Oliver Cromwell. If that is the same Cromwell who oppressed so many catholics in Britan and Ireland in the 17th century, there probably is no tradition of singing of psalms in catholic Ireland. I would love to be proved wrong. I would be among the first in line to get any music.
I wish you luck and I hope to see this discussion continue.

Go n-éirigh an t-adh leat,

Brían.