I research the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For the Hart surname in Co. Armagh, the earliest B rcd is for a Catherine in 1797. There are no I(saac) Hart M rcds in Ireland prior to 1852.
For Mary Carruthers B ????, somewhere in Ireland, with the father's forename unknown, the earliest rcd I found was in 1799, Co. Down. Likely too late for your timeline.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.