Going off something Plon-Plon said OTL, would there be a succession struggle? Plon-Plon hoped that the emperor would die leaving an underage son (who he was always very quick to point out was a dim-witted idiot), since he said: "And then France will rally to me. For they will never accept the rule of a monarchist [Eugenie] and a child [Napoléon IV]" (a variant of this quote is "rule of a woman and an idiot" IIRC).
Not to mention, France's last two regencies (for Henri V, for the Comte de Paris) didn't work out so well. Although, by comparison, in both cases, the king (Charles X/Louis Philippe) was confronted with a crisis and abdicated, rather than that he was dead, as Napoléon III would be here.