Let us just remind ourselves for one moment about the definition of individuation. Jung described it as being a process of psychological integration which resulted in the development of an individual personality. Jung called it the process by which individuals are formed and identify themselves as being different from others, or as "a being distinct from the general, collective psychology."
If we think about this term in relation to the bet, we can see that the lawyer, during his time of confinement, definitely goes through a process where he comes to define himself and his identity in sharp contrast to the rest of humanity. Note what he writes in the letter that he leaves behind, and how he separates himself and his own values and what he stands for from those of the rest of humanity:
You have lost your reason and taken the wrong path. You have taken lies for truth and hideousness for beauty. You would marvel if, owing to strange events of some sort, frogs and lizards suddenly grew on apple and orange trees instead of fruit or if roses began to smell like a sweating horse; so I marvel at you who exchange heaven for earth. I don't want to understand you.
To support his position, the lawyer deliberately reneges on the bet, just a few minutes before he was due to win it, showing his contempt for money and for what the majority count as being precious. He has definitely developed his own character and personality in opposition to the rest of humanity, and thus can be said to have completed the process of individuation that Jung talks about.