Just to confirm Francis & James Hodson were brothers, but James does not seem to have the occupation you describe.
Bury & Norwich Post 29th April 1812
Cambridge – On the 16th inst., aged 58, J.Hodson M.D, of London, brother of Mr Hodson, printer here.
Stamford Mercury 24th April 1812
Cambridge - On Thursday last, Dr Hodson of Cross Street, Hatton Garden, London, brother to Mr Hodson, printer of this place.
Morning Chronicle 18th April 1812
Died – On Thursday last, the 16th instant, Dr James Hodson of Cross Street, Hatton Garden.
Bury & Norwich Post 1st June 1825
On Saturday at her house in Cross Street, Hatton Garden, aged 61. Harriet relict of James Hodson M.D