This situation probably wouldn't have caused that much trouble in regards for the succession to the French throne. France followed Salic Law, so Louis XII would still be succeeded by Francis I in this scenario. Francis' marriage to Claude of France had nothing to do with "easing the succession", even if Louis XII had a pretty bad opinion of his nephew. But he just couldn't do anything to prevent Francis I from succeeding him aside from fathering a son (which is what he failed to do OTL).
The issue could come from Britanny though. Britanny got attached to France thanks to Anne of Britanny's marriages to the French Kings: first Charles VIII and then Louis XII. Anne was still Duchess of Britanny in her own right so when she died OTL, the duchy went to her eldest daughter, Claude. Which is the main reason why Claude was eventually wedded to Francis I: because it made sure the next Duke of Britanny was the King of France. As a matter of fact, the Dauphin Francis (Francis I and Claude's eldest son) was sometimes referred as Duke Francis III of Britanny between the death of his mother in 1524 and his own death in 1536. Same thing with Henri II: he was both titled Dauphin of France and Duke of Britanny before he became King of France in 1547.
If Louis XII and Anne of Britanny have absolutely no children though, then Britanny can technically remain independant since the King of France isn't the next Duke of Britanny. The situation is complex however, as there isn't a clear successor to Anne in 1514. The following candidate I found are as follow:
- François II d'Avaugour (1493-1517) - The son of François Ier d'Avaugour, himself the illegitimate son of Duke Francis II of Britanny. He is the closest kin to Anne, being her nephew. However, he is the son of a bastard, and thus that technically excludes him from the throne. It wouldn't be the first time in history that a bastard might be considered worthy of accessing a title though.
- Philibert de Chalon (1502-1530) - Son of Jean IV of Chalon-Arlay, Prince of Orange. Jean IV himself was the son of Guillaume VII and Catherine of Britanny, sister to Duke Francis II and thus Anne's aunt. He is probably the most legitimate candidate for the Duchy. There are several issues surrounding him though: he is only 12 in 1514 and his father was known as an opponent of France, so that probably won't go well if he is chosen.
- Jacques de Rohan (1478-1527) - son of Jean II, Viscount of Rohan, and Marie of Britanny, herself daughter of Francis I of Britanny as well as the cousin and stepsister to Duke Francis II (her eldest sister Marguerite was Francis II's wife). He belongs to a pretty old and strong Breton family and he has a pretty good lineage to back his claim. His family also happens to have supported France, but his father also tried to claim the Duchy at one point only for Charles VIII to remind him he couldn't do what he wanted.
- Guy XVI de Laval (1476-1531) - son of Jean de Laval, himself son of Guy XV de Laval and Isabelle of Britanny, herself daughter of Duke Jean V of Britanny. He is basically in the same situation as Jacques de Rohan, but he is less well-placed in the succession line. That being said, he apparently played a pretty big role OTL and seems relatively important: he might just have enough political wait to pull it off...
- Charles IV d'Alençon (1489-1525) - He belongs to the House of Valois and is a very distant cousin to Louis XII and Francis I, as his ancestor was the brother of King Philippe VI. He also happens to have connections to the House of Britanny, his great grandfather Jean I d'Alençon having wed Marie of Britanny, daughter of Duke Jean IV. He also happens to be Francis I's stepbrother, as he had married the latter's elder sister, Marguerite of Angoulême (later known as Marguerite of Navarra), in 1509. He is probably the claimant with the least back-up but he is a Valois, a Prince du Sang (He might even be Premier Prince du Sang at the time) and the future King of France's stepbrother...
The last two are probably the ones with the least chance to take the Duchy from my POV, but I don't know enough about Britanny's history to know who would be chosen. If I had to guess, I'd go with Jacques de Rohan because he seems to be in the better position and that would probably satisfy everyone: the Bretons get a Duke from their own ranks and the French have a relatively solid ally. Still, you're going to get a very interesting situation in Britanny in this scenario...