Just had another look at the English side of the Doubles trying to find the James who arrived in the USA to begin the Double dynasty over there. I found a James Double born July 1751 baptized 17 Jul 1751, born in East Grinstead Sussex to a James Double and Elizabeth.(No surname for Elizabeth or date of birth for James though probably born about 20 years earlier).
There's no detail of James b 1751 marrying here or where he ended up. They also had another son Daniel born Feb 1745 and baptized 12 Feb 1745.
20 Years ealier in that place a Richard Double and his wife Mary had two sons Thomas born 1720 baptized 14 Feb 1720 and, Daniel 1722 Baptized 24 Nov 1722. I think it more than likely that they also had a son James born at about the same time who would have become the father of James born 1751. Christian names usual re-occur within families.