I have Hamilton ancestors who lived in Cushendall and surrounding townlands in the 18th and 19th century. Most emigrated to Canada and the US - MA, Montana, VA. They were Catholic then.
I have personally searched the layde/Cushendall parish registers for all Hamilton birth and marriage entries and found many. Of particular note are Alexander Hamilton and Catherine Murray who emigrated to Prescott Co. Canada, and Archibald b.1781 and James Hamilton who emigrated to Grey Co, Canada.
Before Ireland, this line of Hamilton came from Ayshire, Scotland. In the mid 17th century, they left Scotland for Ireland and MA, USA - Brookfield area.
Where do your Hamilton ancestors come from?
A good start would be a dna test which I have done