I don't know if any of this helps but I did a study of all of the versions fairly recently and I can't find the title 'The Unfortunate Rake' attached to any of them.
The Buck's Elegy c1790 (no imprint) has simply 'Covent Garden' as the setting. To me the suggestion that a more fitting setting would be a hospital seems fairly logical for the slightly later versions.

Pearson of Manchester had 'down by the ----------- Hospital' a device where the intention is that the singer fills in his own nearest hospital name. Ross of Newcastle, Williamson of Newcastle, Such of London, all had 'Lock'. These would all be c1850 but the later 'Sailor Cut Down in his Prime' printed by Forth of Hull has 'down by the Royal Albion'

Earlier oral versions from the 1900s of Unfortunate Lass just have 'As I was out walking one midsummer morning' or 'down in false garden'down by the seaside'.

This is only an opinion but the wording of the 1918 Sharp version is more akin to that of the blues song than English oral versions.

I WENT down by St James's Hospital.

It looks like Kidson was the source of the link to the Irish tune
'The Unfortunate rake' in JFSS 5 p254 (1909). Unfortunately the first line of Mrs Thompson's fragment from Knaresborough, Yorkshire runs 'As I was a walking in Rippleton Gardens' probably linking back to 'Covent Garden'. Other Gardens occur in other versions.