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. By 1800, they are nearly non-existant. Your lack of B/M/D data makes the search more difficult and less conclusive.
For ? Caulfield B unknown date, somewhere in Ireland, with a father forenamed Thomas, I found 568 records, with 28 in Co. Wexford. None had a forename of Kate or Cate.