could he parish be St. Bride - rather than St. Bridget ?
This covers parts of Dublin South City - e.g. Bishop Street, Bride St, around Dublin Castle, George's St, Stephen's St etc
I dont see an RC or CofI parish of St. Bridget ... but the Church of Ireland on Bride Street was St. Bridget's, parish of St. Bride.
Some records for this parish are included in the IrishGenealogy website mentioned above - but they only seem to have marriages available.
Shane