The years between the Third restoration and the death of Henri V would be crucial for the survival of the regime. Given the fact of the constant clashes, it is implied that those are going to be difficult times. However, after his death we would see the consolidation of the liberal, constitutional monarchy.
My question goes far beyond: is this regime going to last to the present day? Ceteris paribus, keeping everything except this turning point, including the results of WWI, the French monarchy has fair chances of being kept until the Nazi occupation. The question that intrigues me still is would it survive the Vichy régime? That would depend upon the role of the royals: if they side with them, they would be thrown out (as happened in Italy). If they, instead, rally resistence forces around them, they will be kept until today.