The subscription site Roots Ireland has 17 Patrick Walsh baptismal records in Co. Kilkenny in the 1770s including 3 records in 1778,
There are also a number of parishes in Co. Kilkenny which do not have records back to the 1770s.
So if your Patrick Walsh was from Co. Kilkenny, we don't know if his record is part of the 17 mentioned above or no record is available because the parish he was baptized in no longer has records back to the 1770s.
You don't mention whether ot not you have done DNA testing. Looks like you are eligible to take the Y-DNA test which tracks your pure paternal line (father, grandfather, his father and so on back on the Walsh line). The Y- test will take you back to the 1700s and possibly give you some clues on the county where your Walshs originated.
The other test is the standard DNA test also known as the autosomal test which tracks all of your paternal and maternal lines. From your matches you may get some clues on your Walsh line.