I too have traced the bardy family back to Burton Leonard - John Bardy son George, who we descend from was born in BL in 1777.
two of george's sons moved to York in the 1840's we descend from one of them - the other did have a son called henry who was born 1837/8, ( he is listed as 13 in the 1851 census.
would be good to compare notes.
We too had a family story of French ancestry and Bardy is a commonish name in France - but I don't have any evidence. There is though on the latterday saints site a Bardy marriage in East Yorkshire in the late 1600s!