re St James Hospital/Infirmary (name): if it's indeed a common hospital name (for such ailments anyway), more so than St George or St Paul etc, then that would make the link rather weak. Hmm, are hospitals referred to by name in any ballads that clearly do not have "Unfortunate Rake" related themes?

Well, there is a workhouse:

A boy to me was bound apprentice
Because his parents they were poor
So I took him from St James's Workhouse
All for to sail on the Greenland shore.

(The Cruel Ship's Captain aka The Captain's Apprentice - originally from East Anglia; the workhouse by that name was in Norwich in the early 19th century, I think).