I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 20 M church/civil/census B/M/D records for MOST Irish counties, including Armagh. Records before 1863 are increasingly sparse - almost non-existant by 1800. Protestant records are the LEAST likely to have survived.
For Bri(dget) McLaughlin B 1845 +/- 2 yrs, Co. Armagh, with the parents' named Pat(rick) and Mary, I found ZERO records.
Will leave it to you to search for the parents' B records. Note earlier records are less likely to have survived.