The main problem with the Fourth Republic was not the Constitution itself, but the fact that 2 major parties, the Communists and the Gaullists, were from 1947 onwards excluded from government. As a result, we had an unlikely - and unstable - alliance of the Socialists, the Radicals and the Centre Right (the MRP), and a serie of short-lived cabinets between 1948 and 1958.
By 1958 however, the regime had started to reform itself. Félix Gaillard and Pierre Pflimlin had introduced constitutional reforms designed to make the governments more stable. The Algiers crisis and De Gaulle's return stopped everything, and that was the end of the Fourth Republic.
So basically, for the Fourth Republic to endure, you have to take De Gaulle out of the equation.
This could be a potential POD:
1953: Charles de Gaulle dies of a stroke.
1955: a substential part of the gaullists around Jacques Chaban-Delmas rally to Pierre Mendès France
1956: victory of the Republican Front (more substancial than in OTL), with more seats for the Radical Party than for the SFIO (Socialists). As a result, President René Coty designs Mendès France (and not Guy Mollet) to head the government.
1956 onwards: reforms in Algeria, no Suez expedition, Constitutional reforms to provide more governmental stability, change of the electoral law from proportional representation to two-round majority system.
Probable military coup in Algeria, that ultimately fails due to lack of support from the conscript soldiers. Algerian independence in the early 1960s (as in OTL)
The Fourth Republic emerges from the Algerian crisis as a stronger and more stable regime, comparable to the West German model. No popular election of the President (with all the deplorable consequenses in OTL).