OOC: It really should be pre-1900
I think that, had he survived the assassin's bullet, Louis Napoleon wouldn't have lasted long as President. I mean, Cavaignac won with an impressive conservative vote in 1848, but ended up embroiled in a series of fire-fighting exercises (particularly Blanqui's bloody and stupid attempt in 1850 to overthrow the Government in Paris). By the next election he was so unpopular that he failed to even secure nomination from his own party.
I mean, it wasn't until the 1860s when you finally saw stability in the Republic under the Moderates. Victor Hugo's championing of first Louis Blanc and then Leon Gambetta was crucial in creating the stable liberal governments of the late 50s and 60s.
To answer the question, I've always suspected that Louis Napoleon would have intervened in Italy. Maybe instead of Cavaignac's hands-off approach that saw the Italian Republic we know today emerge, you would have seen French troops suppress Garibaldi.