I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 20 M church, civil, census, etc for MOST Irish counties. Records before 1863 are increasingly sparse and almost non-existant before 1800.
For W(illiam) Stuart B 1833 (exact), somewhere in Ireland, with a father forenamed W(illiam), I found 2 records: 1- Co. Antrim, the other in Co. Monaghan. Both these counties are some distance from Co. Kilkenny. Neither is likely for your ancestor. It may be that the B event wasn't recorded, or the record hasn't survived.