Death
His will (dated 2 Dec 1776) mentions the Hon. Henry Fane of Berners Street, who I think would have been his cousin Henry Fane, (1739-1802), son of Thomas Fane (1701-1777)
who was an MP between 1772 and 1802 and who appears in a portrait with Inigo Jones and Charles Blair (
https://en.wikipedia.org/wiki/Henry_Fane_(1739%E2%80%931802)). Alternatively, it could be his uncle Henry Fane (1703-1777) who was
also an MP (between 1757 and 1777). But I'm less sure that it was the latter person as I don't see that he is connected to Berners Lane.
Later, the will mentions another member of the Fane family:
...to the Hon. Tho. Fane, brother of the present Earl of Westmoreland, & younger son of John the late Earl.
At the time of writing the will (December 1776), the Earl of Westmoreland was
John Fane, 10th Earl of Westmoreland, son of John Fane, 9th Earl. His younger brother Thomas Fane (1760-1807) must be the person referred to by Gilbert Fane Fleming and would have been Gilbert's first cousin once removed.