Robin, the Lock Hospital was (I THINK) at the illustrated site from 1764 to the 1860s, so fits the postulated time ranges.
The only Bodelian broadside with a suggested date (1863-1885) is not reproduced by them. Looking through those available, by the printing I would guess 1850 or later, so in that general range.
Echoing Malcolm, one printer, Masran, used a frame around many sheets that showed blackface minstrel characters on one side. The lyrics could have been about a Scottish lass or whatever. Blackface minstrels were very popular in the British Isles ca. 1850-1900. Any illustration helps catch the eye of the consumer.
I found one reference to "The Trooper Cut Down" (I have lost it, and not the version in the DT) that gave the date 1780 and said it was the ancestor 0f all the Laredo-Pills of Mercury-Unfortunate etc. etc. lyrics. Probably not. The Unfortunates seem to go back to ca. 1800, however (see summary of titles by Malcolm, Penguin Young Girl
Having done some genealogical work, tracing Capt. Townsend is a long, long shot. Royal Naval records are quite good for officers, but not so good for the "foot." The name is not uncommon and is probably invented anyway.