Further to my previous comment. I have just found that the signature of Joseph on his will from 1727 is an older and shaky version of the signature from his Marriage Allegation of 1707, which matches exactly the signature from the administration of the estate of his father, Peregrine, in 1712.
Joseph, son of Peregrine, was mentioned in his uncle Cuthbert Chesman's will, dated 11 July, 1729, which means that the burial of 1728 cannot be that of Joseph of the Bill House.
William