The main problem was the constitution: it provided for a popularly-elected President (universal male suffrage), which opened the doors to the populist candidacy of Louis-Napoléon Bonaparte; for a single-chamber parliament, and for a strict separation between legislative and executive powers, with no way to settle any dispute between the two (no possible dissolution of the Assembly) - no way except of course the coup d'état.
Last but not least, the President was limited to a single 4-year term (he could be re-elected, but after a gap of 4 years).
Of course Bonaparte was elected President. Soon he had to deal with a monarchist/conservative assembly, that was stupid enough to restrict to right of pooer citizens to vote. Bonaparte couldn't call for new elections and he couldn't run again in 1852. So he organized the Coup of December 2nd, 1851, presenting himself as a defender of the people against the oligarchy, and it worked. And a year later he was Napoléon III.
So basically in order for the institutions to function, you'd need either to adopt a copycat of the US Constitution (no term limits for the President, 2 legislative chambers) - I'm not sure it would work in France.
Or you'd need to give the President the right to dissolve the Assembly AND the Assembly the right to censure the Government. In other words, you'd need something close to what is now the Fifth Republic.
You could also have a figurehead President not popularly elected, with the executive power resting mainly on the head of government - in other words, the Third Republic.