Cavaignac would be president until 1852, but a restoration of monarchy is likely after 1852.
The Order Party, a coalition of conservatives, had already the objective of having an Orléans prince (Joinville I believe) to restore the monarchy.
IOTL, they initiated a series of authoritarian laws restricting freedoms.
IOTL, the liberal opposition, the DemocSocs (Democrat-Socialist meaning the actual republicans, not those of the conservative who planned to kill the republic) was relatively powerless since 1848 and the June Days.
Still, given Louis-Napoleon would be dead, a part of the urban and worker vote that went on him could be won by Ledru-Rollin, but not much as desillusion was strong since the repression of the June Days.
Then, Prince Napoleon aka Plonplon would find himself the head of the House of Bonaparte and of the Bonapartist party, but that party would be much reduced as the radicalism of the Prince would not find as much appeal as Louis-Napoleon; under the Second Republic, Prince Napoleon radicalism won him the nickname of the Mountain Prince.
It's likely that with him, we would see an alliance of Bonapartists and Democrat-Socialists; the essential difference between them is that the Bonapartists defend a presidential republic with use of refendum , a kind of 5th like Republic (De Gaulle never admitted it, but the 5th Republic is based on the Bonapartist's Republic idea), and that the Democrat-Socialists are parliamentarists.
I think that Plonplon could be the candidate of the Bonapartist-DemocSoc coalition in 1852.
However, IOTL, early by elections during the Second Republic saw great progress by DemocSocs, progress that scared enough the conservatives so that they vote a law restricting the number of electors by a third to reduce the electoral basis of DemocSocs.
Such thing makes me thinking that the conservatives would still carry 1852.
In my opinion, had Louis-Napoleon not been president and made his coup, the country would have headed towards a new revolution and a civil war.