OTL, after Napoleon III was deposed following the Franco-Prussian War, Bonapartism basically died as a major political force in France. How can Bonapartism continue to be relevant after Napoleon IIIs deposing? What if Napoleon III is more competent and as such has a much less soured legacy, meaning his reign isn't seen as a failed political experiment?
There were opportunities for a Bonapartist Revival. The two that come to mind are the Dreyfus Affair and the Boulanger Crisis that the Third Republic experienced. Boulanger probably could have staged a coup against the Third Republic using his own personal propaganda and restored Napoleon V (Victor-Napoleon) to the throne.
Though for a better chance of success Napoleon IV, the Prince-Imperial should probably survive as well instead of being killed in battle by the Zulus on behalf of the British Army. He probably could had British backing to retake the Imperial Throne during a worse Dreyfus Affair or something along those lines.
What if Napoleon III is more competent and as such has a much less soured legacy, meaning his reign isn't seen as a failed political experiment?
Honestly if Napoleon doesn't lead the French army himself at Sedan and gets captured France probably would have had a much better and more moderate peace since it would actually have an army to resist fighting the Prussians. While this defeat would have been quite humiliating, the Bonapartes probably would have kept the throne and Alsace-Lorraine. Though the Second Empire here would probably be more of a Parliamentary Constitutional Monarchy rather than the Constitutionaloy Absolute Monarchy Napoleon I and Napoleon III (earlier in his reign) ruled with. I'd say the Emperor would have a great deal of executive power however balanced by a more empowered Parliament. Overall it would probably be a much more stable government and 19th Century for France.
As for geopolitical consequences, its likely that France and Russia might ally sooner. France still retaining a monarchy means that ideologically its more agreeable to the Russians. Russia probably would be upset at the balance of power being so altered now that Germany is unified.
As for Italy, I can imagine that the French wouldn't be so happy with them after they annexed the Papacy.
What if Napoleon III is more competent
He was quite competent, but what led up to the Franco Prussian War was that he was basically incapacitated by cancer.
One way could be to have France remain a monarchy. Wanting to put a different person on the throne is still a prospect that gets discussed in modern constitutional monarchies.
The Second French Empire held a referendum about Napoleon III's liberalization which was passed with an overwhelming majority. Its likely that the Bonapartes probably could have had enough popular support to keep the throne had their defeat been less of a clusterfuck.
Napoleon III honestly probably should have stayed in Paris instead of riding with his army to Sedan.