There could totally be ground for bad faith depending on how the negociations went and the intent of the parties, especially if Mbappé implied that his extension would depend on the results of the transfer window and if he made requests regarding future recruitments only to say he wouldn't extend with 6 weeks left in the transfer window. Which actually seems to be the case at least according to PSG that wrote, in its letter to Mbappé :
"Il est vrai que nous avions discuté aussi d’un recrutement très ambitieux que nous n’avons pu que partiellement réaliser compte tenu des conditions du mercato et des contraintes réglementaires françaises et européennes, peut-on également lire. Des conditions complètement hors de notre contrôle, mais en dépit desquelles nous avons essayé de répondre à vos demandes contrairement à ce que nous aurions fait pour tout autre joueur sans être fatal au club."
Now is it a guaranteed winning case? Obviously not but I'd litigate it. Can we rule out gross incompetence? No, "competence" isn't really something PSG has shown in the last couple of years.