I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse and al most non-existant prior to 1800. Many early B/M/D events weren't recorded, or the records didn't survive.
For Cat(herine) Byrne B 1813 +/- 2 yrs, Co. Kildare, with a father forenamed James, mother Mary, I found 0 records.
For Andrew Byrne B 1812 ditto, ditto.
For William Byrne B 1810 ditto, ditto.
For John Byrne B 1805 ditto, ditto.
For Mary Byrne B 1809 ditto, I found 1 record. However, the mother's surname either doesn't begin with S, or wasn't recorded. This is likely a false positive.