Possibly, if he had avoided the FP War
In The Grimaldis of Monaco, Anne Edwards wrote: Louis Napoleon (Napoleon III) was not unaware of the precariousness of his reign. To save the throne for his son, he was planning to convert the regime into a "Liberal Empire" and himself into a constitutional monarch. Would Napoleon III have been successful with a "Liberal Regime" for the French Empire?
Possibly, if he had avoided the FP War. The 2nd Empire was not in the best of shape internally, but absent the shock of the defeat of the Imperial forces, and the siege of Paris, the Bonaparte dynasty might have limped along to the Prince Imperial.
Whether he could have threaded the needle to a parliamentary democracy, and how the various factions within whatever legislature was formed would have lined up for, against, or otherwise in regards to the monarchy, is difficult to discern - lots of potential ripples.
As has been said, there were lots of "doctors without patients, and lawyers without clients (in Paris), all dreaming of something to re-open the gates..."
And even if NIV (presumably) takes the throne after his father's death or abdication (as part of a constitutional monarchy deal), there's always the possibility of some later European conflict breaking open the fissures within French national politics in the late Nineteenth and early Twentieth centuries.
Stable, France was not...
Best,