photo credit: Southampton City Art Gallery
This man is so distinctive that someone might be able to identify him. I think it's a rather good painting and close to Wright of Derby.
My suggestion for the sitter is Henry Harington M. D. (1727–1816):
The collection comments: 'The painting is thought to be a portrait of Nathaniel St André by an unknown artist. It was purchased from the Council of the Hartley (now Southampton) University Library in 1904 by Southampton City Council as a portrait of George Frederick Pitt (donor of the Pitt Library).'
Probably by Mason Chamberlin the elder, see https://artuk.org/discover/artworks/view_as/grid/search/makers:mason-chamberlin-the-elder-17271787
Maybe Sir Isaac Watts. I believe he wrote the music of Southampton Civic Centre bells chime. He has a large nose, which upon looking at other portraits, they also have large noses.
It is also the School hymn for King Edward VI School, Southampton, which Isaac Watts himself attended, the peal of the Southampton Civic Centre clock tower and The Laidlaw Memorial School and Junior College, Ketti. Wikkapeadia
I'm pretty sure Isaac Watts (no 'Sir') was long dead by the time this was painted (?1760s-80s - I agree it could be Chamberlin). There are many images of Watts online, all wearing the bands of a non-conformist minister; and though there is some variability of features, none looks remotely like this man.
Henry Harington is also not possible, Tim, I fear. The portrait on which your print is based was painted by Beach in 1799 (see http://bit.ly/2sbhcVS ), yet shows a far less ravaged face. And there is another c.1815 portrait of him as an old man by Slater (print after it here http://bit.ly/2raSMfK ), exhibited posthumously at the RA in 1819 - this is all too late...and he still doesn't look decrepit enough.
Southampton have rightly concluded that George Frederick Pitt (bap Feb 1766-1835) is too late, and they suggest his natural father Nathaniel St André (d.March 1766 age 96) in old age instead. Portraits of St André (the credulous self-publicist 'rabbit doctor' of 1726) seem to be mainly genetic caricatures, except possibly for this http://bit.ly/2qswDbR ; but even if reliable, it is too early an image to help much. But if the likeness of his supposed son G.F.Pitt in the Royal Collection http://bit.ly/2qptqOk (and see also http://bit.ly/2qwns9u ) is authentic, his aquiline nose and protuberant chin (even allowing for some exaggeration by Dighton) seem unlikely to have emerged from the genes of the man in our portrait. But perhaps the ninety-something year-old Dr St André was as gullible about being the father of his former maid servant Mary Pitt's two bastard sons as he had been 40 years before about Mary Toft being the mother of 15 rabbits. His death shortly after George's baptism meant that he did not live to see some rather surprising facial features grow from his alleged son and co-heir's infant face.
So on balance I feel Southampton's suggestion is still very plausible - certainly it seems to have come from St André's collection, left to Southampton by Pitt; but to prove it will probably be impossible.