If he was described as a pauper in the 1851 census, this almost certainly means that he was receiving relief from the Parochial Board, the committee responsible for administering the Poor Law after 1845. If so, the Board's records would have collected a lot of detail about his family circumstances, and would have noted his death when it occurred.
The big question is whether or not these records have survived. If they have, they should be in the care of Ayrshire Archives
http://www.ayrshirearchives.org.uk/ or if not the Archives should know where they are. So I suggest that you contact them.