Hi Tom
Wondering where in the world you are? I have looked in Australia for records, only because I have seen Dwyer/Hourahan families here.... so not really a valid reason for looking. But a little out of the box thinking....
Wondering if your Dwyer was military? Reason I ask is IF the Dwyer father was military, the births may not be in the normal locations like parish registers (should be but may not be)... they could be listed in the British Army Births, which are on Find MY Past. That applies to births worldwide, including Ireland. I can see a lot of Dwyers born in Australia to military men (1788-1850s), their registration clearly says British Army Births (or British births overseas/military etc) These men served here during our convict era and whilst we were a colony/several colonies... then returned to Ireland or England when their service ended. They DID bring their wife and children with them, depending where they served. We generally know our ancestors lived in specific locations and for the military, we say they lived in Ireland, but served overseas... but some of the overseas services saw births and baptisms to their wife. With the education back then, the children could have believed they were born in Ireland because that is all they remember.
Hope that helps.... if he IS military.
Always happy to help from this side of the world if there are records needed, just ask.
Cheers.