The story of "20,000 pounds going to Chancery" came from the family in England. Eric Wolfe, a descendant of Eliza Bickerstaff's in England, corresponded for a time with Bob Bickerstaff, a grandson of Thomas Bickerstaff's here in the U.S. [Eric and Bob would have been third cousins.; Eric was in his 80's when he tracked down the Bickerstaffs in the U.S.] It was he who passed this information to Bob.
As far as I know, the only info passed down directly from Agnes or Thomas was the "little black book" Agnes compiled on the family history. Although my mother believes she saw it as a child as her father Richard had a copy, no-one knows what became of this book And which version of the Bernard/Burnard story did it contain? Supposedly, at first Agnes believed their father was an Henri Bernard, from France, and Hannah was a dancer who would not give up dancing to accompany him back to France. [The only occupation I've seen in England censuses showed her to be a lace maker.]
In the 1900 U.S. Federal Census, both Thomas and Agnes stated their father was born in France; in 1910 Thomas said the same [Agnes has not been found in that census.] In 1920 both stated their father was born in England. Agnes' death certificate  showed father's place of birth as "Tames", England, name unknown. Thomas's death certificate  showed father's place of birth as England, name unknown. Agnes's return voyage from England and probate court was April 1913; evidently she learned the name of their father between 1910-1913. Hannah's sisters Eliza [Kirtland] and Elizabeth [Bateman] both died before 1900 so it was not they who imparted the information to Agnes. Although Agnes' correspondence was with the Batemans [grandchildren of Elizabeth], it was Eliza's great-grandson, John Eric Wolfe, who conducted the U.S. correspondence and reported on the probate matter to Bob Bickerstaff. It seems the whole family contained clues to the father of Thomas and Agnes; I wonder why they waited until after 1910 to pass it on?!