I've found a reference in the D'Arcy Registered Pedigree (Vol. 14, image 176) at the National Library of Ireland to "Sir James Buchannan, Bart." Since his granddaughter was born in the late 1700s, he was probably a baronet in the early- to mid-1700s. But Baronet of what, or where, is proving to be difficult. I did come across a news clipping in the 23 April 1777 Saunders's News-Letter in Dublin (p.1) mentioning Sir James Buchannan, bart. who sailed in the Hillsborough Packet for Holyhead. (So I at least know he exists!)
I've been working my way through lineage books all morning, but am not having much luck. He may have been born in Scotland, but he was certainly in Ireland, and probably Dublin, by the late 1700s.
I've attached this to his granddaughter's profile, since his own information is still so tenuous. If anyone has any idea who he might be, I'd love a clue or two.