Sharon:
The 1853 Griffiths Valuation head of household listing has a Michael Briggs in Burrow townland in Rosslare civil parish. https://www.failteromhat.com/griffiths/wexford/rosslare.php
I located the death records for Michael and Anastasia in 1874 and 1884 respectively. They were living in Rosslare. The informant information was hard to read. For Michael, the informant was Jane Briggs but I can't tell how she was related. Was she a sister of James? Maybe it is indicating James?
eighth record for Michael's Briggs' death reord December 30 1874 https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
third record for Anastasia's 1884 death record https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
The RC parish for that area is Tagoat and marriage records start in 1853. I was not able to find the marriage record or a baptismal record for James on the subscription site Roots Ireland.
I did locate an 1876 civil marriage record for Maria Briggs sister to James see third record https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_re…
Also found another marriage record for a sister Jane Briggs in 1878 see fourth record https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_re…
so that explains the informant for Michael's death
Roger McDonnell