Mary Brien indicated on her civil marriage record that she was 23 in 1876. Frequently marriage ages were understated but we have to assume that 23 was reasonably correct. The subscription site Roots Ireland only has a partial list of Co. Wexford parishes in their data base- less than 50% so I usually use FindMyPast (subscription) which covers all parishes in Wexford. The only record i could find was a January 21 1851 baptismal record for a Mary Breen (as transcribed) in the Ballymore RC parish with father Patrick Breen and mother Catherine Treacy. There was no record in Tagoat parish.
Ballymore RC parish is immediately to the west of Tagoat RC parish. I then tried to find a marriage record with no success.
I also note that the 1853 Griffiths Valuation head of household listing for Rosslare civil parish shows a Patrick Bryan in Burrow townland which is the same townland for the Briggs family. Is this your Patrick Brien or someone else?
You indicated having a few leads for Patrick and Catherine. Can you share?