Shine:
The free site irishgenealogy.ie (Church Record Tab) has Dublin city baptismal records. I see three in 1819 (two Roman Catholic and the other Church of Ireland), There is also an Andrew Byrne baptismal record in 1820 and another in 1815. If you have the names of his siters, you may be able to narrow down the possibilities.
We have the same constraint with Catherine Caragher (lack of parental names). The subscription site Roots Ireland shows one Catherine Caragher baptismal record in 1833 in Co. Meath but there are also Catherine Carragher records in 1831, 1832 and 1834 in Co. Louth and Co.Armagh.
Have you considered a DNA test. You may have matches with fourth cousins who may have more info on the origins of the family in Ireland. You may also want to search on Ancestry.com You may get some clues. Possibly someone has a family tree with your ancestors.
Roger McDonnell