Louise.
It was Archibald's occupation I was after, to look for possible deaths in Ireland. However if he died before 1855 then there won't be a statutory death certificate as they hadn't started in Ireland then.
Marriage registration started in Ireland in 1845 (1864 for RC marriages) and birth & death started in 1864. Prior to that we have to rely on church records, where they still exist. And many of those early records are not on-line, especially for Protestant denominations. Clearly Archibald and Janet married and had children long before the start of any of those statutory records. Where church records do exist, they are mostly held in PRONI (the public record office) in Belfast. Personal visit required to view them.
Kane is one of those surnames which can be various denominations. Plenty of RC but plenty of Presbyterians and Church of Ireland (Anglican) too.
Going by the 1901 census, Sinclair, in Co Antrim, is mainly Presbyterian and Church of Ireland but with some RC too. Laverty is similar but with perhaps a greater proportion of RC.
Did Peter and his wife have any daughters? If so, what denomination did they marry in? That may indicate his own denomination (and therefore the relevant denomination to search in Ireland). Not all churches have baptism and marriage records for the early 1800s and so there is a possibility that there are simply no records to find.
You have searched the 1831 census for Co Derry. It's more or less the only bit of the 1831 Irish census to have survived. It had been removed from the Public Record Office in Dublin, for some reason, when the building was burned in 1922 during the civil war, and the rest of that census largely lost. So there's no 1831 census of Antrim to search for this family.
Did the family ever claim under the Poor Law in Glasgow, do you know? If they did the Poor Law records are in the Mitchell Library and usually record the applicant's parish of origin in Ireland (with a view to possibly returning them there). The Mitchell will search those records for you, for a small fee.
You are lucky to have an 1855 Scottish death certificate. That year only, the records contained a lot more information than in later years.
If Archibald was a labourer, same as his son Peter, and died in Ireland then he is unlikely to have a gravestone. Few labourers could afford one. Only the Church of Ireland kept burial records. Most other denominations did not bother. So a death in Ireland pre 1855 may be hard to trace. And of course death registration only started in Scotland in 1855, and so tracing a pre 1855 death there may be challenging too. Scottish church burial records are very patchy, to say the least.
Moving to Campbeltown is a hint that the Kane family may originate in the north east corner of Co Antrim. It's right opposite the Kintyre peninsula. Just 11 miles at the closest point. Folk from the Glens of Antrim went across to Campbeltown and the Kintyre peninsula all the time for work and fairs, and many settled there. Going right back to the 1600s and earlier, many of the families in that part of Antrim originated in Kintyre, and moved to Ireland either for reasons of persecution eg the McAllisters who were pursued by the Campbells, or for economic reasons. So there was always a huge amount of going to and fro. There used to be a regular passenger ferry from Cushendun to Southend on Kintyre, and that's how they mostly got there. (You can still do it today in the summer months, but from Ballycastle to Campbeltown now).