My take on the significance of the “5th of September” movement is that in France, it is the date popularly associated with the beginning of the Reign of Terror. It is the date when Bertrand Barère, a member of the Committee of Public Safety, made a speech supporting it, and ended with the exclamation, “Let’s make terror the order of the day!” From this, I would guess that the prisoner is, or is pretending to be, a member of a French insurrectionist organization, probably one that seeks to restore French rule on St. Croix and/or St. Anne through political violence.
Given the limited amount of information we are given in the novellas, I don’t think it is possible to fully understand the nature of the political unrest that is the background of the 3 novellas, only that St. Croix and St. Anne appear to be in a state of armed hostility, if not quite war, and that the descendants of the French settlers and the anglophone settlers seem to be continuing their war. It’s not at all clear if the government on St. Anne (where we are told the original French settlers were defeated in a fiery battle) is a different group of terran settlers than the reigning faction on St. Croix - some of whom (on each planet) may be Abos, who are continuing the conflict between the Marsh People (controlling the government and secret police) and the Hill People (representing the criminal underclass), as I argued in another post. And/Or, the Abos could have assumed the political conflicts of the settlers they killed and mimicked, just as they assumed their features.
Wolfe, I suspect, doesn’t make it clear exactly what is going on in these decadent backwater Earth colonies, because he wanted to create a sinister, murky, and hostile climate for the action of the novellas to take place, like an otherworldly version of the settings in Graham Greene’s novels, with paranoid bureaucracies and natives whose motives are often opaque. The nature of the political setting may be as unimportant as the nature of the unrest in "Paul's Treehouse". (Although Greene's writing style and Wolfe's differ markedly, and their politics grew apart as they grew older, they share quite a bit in terms of theme - both are Catholic converts (who converted upon marriage to Catholic wives), and a Catholic worldview is pervasive in both writer's works. I have to think that Wolfe was probably familiar with Greene. Fifth Head was published in 1972, and it's likely that Wolfe had read Greene's works "The Quiet American" (1955) "Our Man in Havana" (1959) and The Comedians (1966) (or seen the film adaptations) - all were immensely popular and feature settings with inept but sinister security bureaucracies and corrupt backwater, third world settings that might have influenced Wolfe's depictions of St. Anne and St. Croix.