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The 1861 UK Census records show that he was born in Eyam, Derbyshire in 1785.
He married Sarah Andrews at St Andrew, Holborn in 1815. He was from that parish.
A christening record in 1815 at St Andrew, Holborn for son Edward Andrews shows his parents address as Warwick (Cot); his father was a cutler.
He entered his only mark at Goldsmiths Hall in 1822 as a plateworker from 104 Hatton Garden, notifying a change to number 27 later the same year.
Christening records at St Andrew, Holborn 1818-32 for 7 more of their children show their father as a silversmith at Hatton Garden.
The 1851 UK Census records him at 1 Southampton Street, Bloomsbury as a retired silversmith and at the same address in 1861 he had become secretary of the Goldsmiths Benevolent Institution.
He was an inmate of the District Suttons Hospital, Charterhouse in 1871 and an entry in the Holborn register for an Edward Barton aged 87 years suggests he died in 1872.

As a fair possibility for the subject gentleman, there's an Edward Barton listed in the 1841 UK Census in Milton, Gravesend, Kent who is about the right age and has seven children that are twenty years of age and under. So far as I can tell, none of the family were born in Kent. His wife's name is Sarah, and the twenty-year old son Alfred's occupation is silversmith. Apparently, Edward has an independent income?

Some details of Edward Barton's son, Alfred, can be found here:

Alfred Barton