Bernard’s birth is long before the statutory records for birth began (1864). To have any chance of tracing his birth in Ireland you need more information. Ideally you need to know his religion, parish and townland in Ireland. You may be able to get that from Scottish census records (check every census), obits and wills, or Poor Law applications (kept in the Mitchell Library, Glasgow). If he died or married in Scotland the relevant certificates will have his parents names which will help.
Few Irish records exist prior to 1800 and even around 1821 they are very patchy, especially for RC, which he seems likely to have been. There may be no record of him for you to find.