The James Liddell born in Stepney on 29? Sept 1765 doesn't seem very likely to be our JL.
[I say 29? because I have a note of 28 Sept, worked out from the christening record giving his age as 15 days at christening on 13 October.]
That would make him 73 at death on 13 July 1839, not 74 as given in the West Briton on 19 July 1839 (and his family are generally accurate in giving their ages in e.g. censuses).
His father being a carpenter, and mother unable to write her name at marriage (31 Aug 1760 St Botolph Algate, London) is of course possible, but isn't a likely background for a stationer and printer.
Thirdly, the biography of his son James, which is generally accurate, refers to his parents as being from Lancashire; now that is certainly wrong as regards his mother, Mary Martyn of St Columb Minor, but seems an unlikely mistake if he came from London. My guess is it might be a mistake for Lanarkshire, a likely area for Liddells, whereas I can't find any Liddells from 18th C Lancashire.