Louis-Philippe used the style Louis-Philippe, Roi des Français. This neutral, sleek, modern royal style is identical to that of his son-in-law, Leopold, Roi des Belges.
With a more traditional territorial designation, but also without any claim to rule by divine grace, this style has also been adopted by the modern Norwegian, Swedish and Spanish monarchies.
Napoleon III on the other hand, used the style Napoléon, par la grâce de Dieu et la volonté nationale, Empereur des Français. Highly ambiguous, just like his empire!
Of course he could have claimed this divine right when "the will of the nation" deposed him (as the National Assembly formally did on the 1st of March 1871), but it would look very strange for a ruler who had come to the throne through elections and plebiscites.
BTW according to the republican-imperial constitution of 1852, the next heir after his direct agnatic descendants would have to be an adopted collateral member of the Bonaparte clan, and as far as I can see no such adoption was carried out. Thus "legitimist Napoelonic rule" must be considered to have ended with his childless son's death in 1879.
