French monarchism was more or less discredited after the 19th century, though it experienced a bit of revivalism in the interwar period with some far-right groups.
It didn't really experienced a revival : if something, it was more rivaled by other far-right, either traditionalist or fascist. Basically, there's not much difference between AF in the 1910's and 1930's.
How about Bonapartism? Was it gone for good after Louis Napoleon?
Strictly speaking? Yes. Bonapartist deputés were eventually merging with center-right moderate republicans, as Orléanists did.
Maybe in French political culture, Bonapartism is just one manifestation of following a strong general. Boulanger and de Gaulle are other examples.
According René Rémond, who produced in his "Right-Wing in France since 1830", Bonapartism (conservatism, charismatic regime, autority) is one of the three basic origins of current far-right with Legitimism (anti-revolutionnary, traditionalist, clerical and somehow paternalist), Orleanism (liberalism, economic).
It's more than just a strong general/strong man appearing, but a conservatism build with state interventionism, representative of a part of french right-wing : De Gaulle, but as well La Rocque and the PSF.
Boulanger....is a special case : his political base was far far more mixed, and went from far-left to far-right, because everyone saw something different, while Boulanger himself didn't have a real program. I think that calling it a populism would be better fit.
I mean, if Legitimists and Orleanists didn't die out completely until after WWII or so, why not Bonapartists?
But they died out completly. Safe internet groups and whatever remains of AF (and let's be clear that even other far-right movements points and laugh at them)...I mean what remains is folklorism without any influence whatsoever.
What exactly is the political ideology behind Bonapartism?
It depends a lot on what we call Bonapartism : but roughly (if we're talking about Napoleonic reign), we have a charismatic regime, legitimized by a direct link with the nation (trough referunda) and above institutional representation (while directly intervening on institution when so fit). A mix of strong man and non-representative republicanism. It's eventually quite proteiform, even during a same reign (see how the IInd Empire changed between 1850's and 1860's).
On a broader sense, when we talk about "Bonapartist right-wing", it's the successor of these ideas, you can have in Gaullism : direct and "personal" relationship with the nation (see the large use of referundum by De Gaulle), opposition to a parlementarian-led institution in favour of an executive-led, state percieved as a mean to apply a policy, political interventionism.
Because that's sorta what the French Empire was initially about, wasn't it?
It's more complex : eventually the Consulate was more a result of an anti-parlementarian factionalism (with Seyies, for exemple) led against revolutionaries elements as Jacobins and Neo-Jacobins while keeping in most structural changes since 1789.
It's more "strike at the left, then strike at the right" in order to reach a stable, strong governemental power and prevent a certain conception of republican regime to be challenged.
Yes, it would be a weird mishmash ideology, but then so was fascism if you think about it.
It would be a bit going against what Bonapartism was IOTL : especially the "above parties and factions" part. You can see that Louis-Napoléon's victory depended not only of his name, but as well (and maybe more) from the fact he never really depended from one or several factions, appeared as a "new and clean man".