William Martin was the first Covenanter (Reformed Presbyterian) Minister in Ireland. He was ordained in July 1757. So your ancestor James Morton (and any siblings he had also born in Ireland) would likely have been baptised into the Presbyterian church (as opposed to the Reformed Presbyterian Church) because that was the only Presbyterian church in existence at that time.
Rev Martin was the only Covenanter Minister in Ireland for quite a few years. By the time he left for the Carolinas there were still only 4 or 5 Covenanter Ministers in Ireland. 2 of them died and 2 left for the US. So it was a very small church in the mid 1700s, and as far as I am aware they didn’t have any permanent church buildings. Services were held either in someone’s barn or in the open air. This hasn’t helped with record keeping!
It was the English King that the Covenanters and the other Presbyterians were in conflict with in the 1600s (known as the Killing Years), and that led to a lot fleeing in both directions across the Irish Sea depending on where the authorities were pressing them most, at any given time. The conflict wasn’t particularly with the RC church, who were also subject to various forms of persecution (the Penal Laws, as they were known), but with the English King and the Church of England (which is Episcopalian).