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 sparse.
For Mary Barkley B 1804 in Co. Meath, I found 0 B rcds. At this early date, less than 1/2 of its parishes were rcd'ng B/M/D data. It may be this B event wasn't rcd'd, or rcds didn't survive. There are no notes which indicate that more rcds may be added.
Suggest you google the Foundation's name to lead you to the database, register, and browse.