To turn the question around a bit, what it would have taken for the Paris Commune to even survive would have been for its revolutionary inspiration to spread. Certainly it would have had to trigger other urban Communes in France's major cities and towns; it would also have had to inspire some more or less compatible movement among country people, otherwise Paris and all the other Communes would simply starve.
But if the Communard movement could get that kind of traction in France, what would stop its inspiration from overturning the established order in many places outside of France? In the Lowlands, in Germany, in Britain...
This of course is the very thing the rest of Europe, or more precisely, the consensus among the propertied ruling classes of Europe, including French ones outside of Paris (and presumably some in Paris, who either fled or kept their heads down and mouths shut for the moment)--this is what they feared. This is why the PC was so many light-years from a prospect of long-term success. It's not that they certainly could not work out their organizational problems--we don't know, the forces of reaction were quickly mustered to crush it, precisely because it just might have worked.
So--a Paris Commune that survives at all is a Paris Commune that has touched off revolutionary civil war, not just in France but throughout western Europe, and who knows maybe even the United States! Assuming the Communard movement prevails, there's no reason to assume it does so only within the boundaries of France and every likelihood all the international borders get redrawn--in fact that the nation-state gets abolished in favor of some federated hierarchy of city-states. If they prevail that is.
Or one can imagine that this wave of revolution does prevail in some places and not others and the upshot is two armed camps glaring at each other across the smoldering ruins of Europe; it would take a detailed timeline (one that has to cleverly and plausibly solve a lot of problems the PC faced that make it hard to argue with the "ASB" dismissal, much as I hate that sort of blanket labeling on principle!) to suggest where the strongholds of which sides would be, where zones of contention would lie, what sort of polarizations and reconfigurations the long hard civil war would catalyze, and so on.