I don't think that the coup plot being discovered would mean Louis-Napoléon overthrown.
You have to get the context. Since the June Days and the conservative takeover of the Assembly, the Party of Order went on huge restriction of freedom, and with the law that removed those who couldn't give evidence of three years residence, which removed one third of the electors with most being poor and low income classes, you had social tension building up.
I may be exagerating, but it seems that if the coup hadn't happened, the country would have headed straight to another Revolution and civil war.
In this context, the consensus in the military and some financial circles was that the Prince-President was the only one able to funel that anger and avoid explosion while federating contradictory components of the socio political landscape, due to his social populist tones (worker classes) and name recognition (countryside). So, as it matters the coup plotting, the army was on the side of the president.
Also, there is that feat that as he has been elected by the people, courtesy of direct suffrage, the President can argue of an equal legitimacy to that of the National Assembly, and that institutional antagonism was one of the reasons that doomed the Second Republic from the onset.
So, to answer the question of knowing what a direct confrontation between the Assembly and the President would be in case of the plot being discovered, it's still very likely that Louis-Napoléon would still win the contest. The difference is that it would go less smoothly as IOTL, the president being forced to react quickly to preempt an impeachment by the National Assembly, resulting in more violence and victims, but still a successful coup.