Hmm...perhaps the Albigensian Crusade is defeated and in the aftermath of this defeat, the Cathari movement grows beyond the ability of the Count of Toulouse to control?
Or perhaps the Count of Toulouse (or some Cathari nobleman if the CoT wasn't Cathar) becomes King of France in the aftermath of the failure of this crusade?
Firstly the Cathars were under attack by the the proto-Dominican Friars before the crusade began, and the Cathars were protected by secular and spiritual lords of southern France as a tool to prevent control by the King of France and the Pope.
Secondly the crusade suffered from several set backs during its course including Count Raymond retaking Toulouse from the crusaders several times. The commander of the Crusaders Simon de Montfort, father of Simon de Montfort, earl of Leicester, the bane of Henry III of England was a very capable man. A master military strategist de Montfort was killed by a chance hit by a stone thrown from the walls of Toulouse in 1218, and the command of the crusade fell to his less capable son Amaury.
King Louis VIII then took up command of the crusade and the crusade dragged out and lasted 20 years and resulted in the Dominican Friars and Papal Inquisition, but there was violence for almost fifty years after.
Fourthly at this time the title of King of France was no longer elective after the twelfth century and the sole dominion of the Capetian family, as the Kings would crown their heir as King during their lifetime.
Maybe if the Papal legate Pierre de Castelnau isn't murdered, or if King John of England is successful against France and Phillipe II looses the Battle of Bouvines then the Cathars may survive a bit longer. Just my two cents.