Likely as a moderate dictatorship under Cavaignac. The Constitution of 1848 allowed for a single term of four years, and as soon as that's up the monarchists (Legitimists & Orléanists), the republicans (moderates & socialists), and the remaining Bonapartists are going to take the opportunity to try and seize the reigns of power. Louis-Napoléon wasn't the only, or the most charismatic, Bonaparte in the French government following the 1848 Revolutions, and nor was he the one with the most, or best, allies in and out of government. Plon-Plon will likely make a bid for power, and so might Pierre-Napoléon. The monarchists will likely have united around Henry, Count of Chambord, with Ferdinand Philippe, Duke of Orléans being the driving force behind a potential Legitimists-Orléanists (re)union. Instead of actively running for presidential office the monarchists will likely, as they did IOTL, attempt to dominate the parliament so that they can simply offer the crown to their chosen candidate. The republicans, especially the radicals, are going to take the opportunity to strive for their demands, both legally and extra-legally. You'd likely see a repeat of the June Days, and there will be even stronger calls in 1852 than there were in 1848 for Cavaignac to simply seize power and declare himself Dictateur; if things get hairy enough and it looks like the whole situation will collapse otherwise Cavaignac would do so. However he doesn't have much longer to live anyhow, assuming he isn't assassinated. IOTL he died in 1855, imprisoned since the coup d'état of 2 December 1851, so he'll probably have a few more years on him; however even assuming he lives longer the situation will become very chaotic without a strongman keeping order. A conservative republican whose willing to get his hands dirty, such as Theirs, might be able to pull something out of the situation, especially if Cavaignac had tapped them as a successor prior to his death, but it's going to be very-touch-and-go for a while.
And that's only in France itself. Several issue that would be important to French politics would actually take place outside of the country, such as the Italian Cause and the Roman Question, or French relations with her neighbors (read: Britain, potentially Austria, less potentially Prussia).