I searched the Irish Family History Foundation's database for Louth. Rcds B4 1840 are sparse. I think it only has RC rcds.
1. Your Callinford is likely Carlingford parish.
2. For Hugh (+variants) Rice (+variants), the earliest B rcd is in 1824
3. For Mary Welsh (+variants), the closest B rcds are dated 1797 & 1796. One of these MAY be for your ancestor. Your chances at this early a date are abt 25%.
4. For a marriage of Hugh and Mary, the earliest rcd'd M date for Hugh is 1836 in Dundalk parish. There are 3 M rcds for Mary in 1836, 2 in Clogerhead and 1 in Dunleer, but none in Dundalk. Therefore, the rcd for Hugh is likely a false positive, or there is a transcription error.
5. Offspring: Barney is likely a pet name for Bernard. There are 0 B rcds for Barney, but 8 B rcds for Bernard (+variants) between 1824 & 1844. For James, ditto, there are 22 B rcds. There was also a priest who liked to use abbreviations rather than the correct forename.
6. To validate a B rcd, you will have to view the detailed rcd to see if it shows the right parents' names. Start with Bernard. If you validate a rcd for Bernard, the birth parish will likely be the same for James.
Google the Foundation's name to lead you to the database. Pls post the results of your search in a reply to this msg on Genforum.