Well, today I heard back from the source of that information, Nicholas Harding (who sprouts from the same family tree, apparently?) with more information, in the form of an obituary confirming that indeed, a Dr. John Perkins of Brussels, son of a John Perkins of Southwark, indeed died in Old Guyana in July 1818. (Apparently another of his correspondents turned up the obituary on a lucky Google Books search.)
This led me to attempt a second search, which turned up gold! That is, it turned up a book titled A Narrative of the Expedition to the Rivers Orinoco and Apure in South America by G. Hippisley (1819), which includes a lot of detail regarding the fate of Dr. John Perkins, Esq., of Brussels–written by someone who traveled with him during his last days. The book discusses in passing Perkins’ service under General Bermudez in the Venezuelan War of Independence, and provides details of his death. Apparently Perkins regretted joining up with Bolívar, who reportedly screwed him over in the end and left him penniless; the poor fellow was trying to get back to Europe when he died.
Moreover, by that point Perkins had a female traveling companion (whom the author, one G. Hippisley, describes as a “petite chere amie”) who had apparently had been Perkins all the way back to Brussels. This was not the Ms. Cook mentioned in Harding’s earlier source (supposedly the Perkins family governess, with whom John Jr. supposedly eloped) but rather a Flemish woman named Eugenia. What happened to that Mrs. Perkins, I cannot say, nor whether Perkins was by then a widower and Eugenia the successor to his affections; the book seems not to dwell on the question, though it seems to me to attempt to make it clear by its innuendo that they were, at least, not married. (Hippisley would surely just call Eugenia the man’s wife, and not his “petite chere amie”, were they married.)
(Eugenia also sounds like a real tough cookie, as Hippisley observed: wading through rivers in the bush and so on.)
Ultimately, John Perkins Jr. succumbed to an unexplained fever, by the way; there is mention of a detailed journal of his trip and of the campaign, though its ultimate fate was unknown… the author speculates that it went to Martinique with Eugenia, but he doesn’t know for sure.
Other little bits and ends come up; Perkins’ subscription to The Medical-Chirurgical Review and Medical Science Quarterly, for example, seemed not to have run out even by 1821–he’s listed among the subscribers for both 1820 and 1821, the former listing him at Rue d’Orangerie in Brussels. There’s also some mention in one genealogical survey (one of Burke’s unavoidable genealogies of landed gentry, the first hit on this page), which seems to confirm Perkins did marry a Caroline Cook, and at least one child, and died in South America, though I cannot access the entry directly.
Anyway, it’s possible Hippisley’s book will yield more information: I’ve only skimmed it. But quite a life!