It could be that he becomes a martyr and instead of a republic the nation rallies around the Empress and her son, either fighting on in a useless spurt of patriotic fervor, or sullenly accepting the peace she proposes.
He would certainly become a martyr, and most likely would end up resting in Les Invalides, maybe side-by-side with his more famous uncle. Incidentally, recently the French government has become considering bringing Napoleon III's body home from his tomb in England.
There would probably be also a rally around the prince imperial, but the empress might be a liability, given her conservative and ultra-catholic views which would contrast with the necessity of implementing some liberal reforms to avoid riots and insurrections in Paris and Lyon (IOTL as soon as the news of Sedan arrived, Parisian mobs started to agitate and the empress - who acted as regent in the absence of the emperor - was immediately abandoned by all factions. She left Paris on the 4th of September IIRC, and managed to reach the Channel ports and subsequently go in exile to England only through the assistance of the Italian ambassador (count Nigra) and the Austrian one (prince Metternich).
It is clear that the deeply conservative government of count Palikao and his faction of Ultramontanes and "Mamelukes" has to go and be replaced by a government with support among the liberals and the moderate left. Maybe Ollivier could become prime minister again, he was a man for all seasons but he had some liberal credentials.
The big issue is the regency council, and I believe that the empress will have to step down. The best bet might be prince Napoleon-Joseph Bonaparte (Plon-plon), son of Jerome Bonaparte and with strong liberal credentials. Plon-plon becomes the head of the Bonaparte house upon the death of the emperor, and is married to a daughter of the king of Italy. The problem is that Eugenie hates him, so she must step down.
On the diplomatic side the death of Napoleon on the battlefield would send ripples all across Europe, and probably the best bet for the new regency council would be to try and involve other European Powers for a mediation between France and the NCG. There should be a lot of worries in the European chancelleries: after the war of 1866 Prussian annexations in North Germany met the displeasure of Russia (that was why Bismarck avoided similar annexations in southern Germany and went for a mild peace treaty with Austria). The possibility of a complete French collapse and subsequent Prussian gains in north-west France would not be accepted. A conference of Powers - called for by Austria and Italy and backed by UK and Russia - might become a real possibility ITTL.