I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds B4 1840 are increasingly sparse, rare B4 1800.
For Mary Mooney B 1860 +/- 4 yrs in Co. Leitrim, I found 1 B rcd in 1863. 16 of 24 RC parishes were keeping B rcds by 1860. However, this rcd has a father named Francis. Using a broader search range, I find two B rcds with a father named William in 1867.
Google the Foundation's name to lead you to this database. Pls post your search results to my genforum msg.