Actually, Jonathan, the matter was almost as embarrassing for the French since they set up the Republic and then abandoned it. The general commanding Fort St Elmo capitulated without making any attempt to insist on protection for collaborators. English writers, going back to Southey, I believe, have largely considered the fate of the Neapolitan Jacobins as a blot on Nelson's reputation.
As for Caracciolo, since he was rather obviously guilty of treason, the most one can say was that he should have had a proper trial - but it would necessarily have been by Royalist officers - and been shot rather than hanged.
The whole history of the Neapolitan Republic is depressing and unsavoury. Captain Troubridge had been sailing round the islands with a set of tame Neapolitan lawyers hanging anyone who had accepted office under the Republic.

regards,

Susan