Whats the question exactly?
The faulty logic that the given sentence contains is B. Either-or Fallacy. An either-or fallacy is the limiting of possible answers into two or it is oversimplifying. In the given sentence above, it only gives two possible answers: either meeting the deadline, or if not the town will suffer.
2. The whole Nazi environment probably aided to these Sonderkommandos becoming collaborators, from seeing everyone around you either fleed, killed or tortured is enough to make people to make choices like that. It was a live or die situation and under those circumstances, anything could happen.
3. Levi and Langer want us to see this situation from the Sonderkommandos point of view, where virtue and morality don't apply and the only choice is to kill/ follow the Nazi orders or be killed. They probably thought that by taking on this task there was a chance of survival.
4. The choices that we make help us to develope as person and in our personal identity, and in the absence of meaningful choices it can confuse us in our identities, and can lead us to make critical descisions.
Answer: The miller gets a job as the town's best standup comedian
