Mortimer asks him to not wear his mask on purpose so people won't know what is behind the masks of the real society members.
No matter the precautions you set up before you talk to Chauncey, the assassin still kills him at the end of dialogue. As soon as Chauncey closes dialogue, the steam room door opens. Quickly closing it causes the initial attack from the assassin to fail, but after a few seconds the assassin will simply shoot him through the door. Even attacking and killing the assassin before this second attempt will simply result in Chauncey dropping dead from a scripted effect. This makes it impossible to save Chauncey.
It is possible to advance in Beyond the Beef without getting Chauncey assassinated.
"Those are all members of the White Glove Society. Our founder, Marjorie, gave us all a dress code. There's only one rule to it. In her words, we must dress in such a way that no one can be said to have dressed better than us."
"I don't know exactly. I wasn't in on it. I think some of them have stopped trusting me. But you can bet they're keeping him near the Gourmand. Our chef, Philippe, has an obsession with fresh ingredients."
pcxbox360 Targeting Chauncey during the quest "Beyond the Beef" can cause a crash on PC. And he can die in the pool of the hotel. [verified]
pc Sometimes just talking to Chauncey in the steam room during the quest "Beyond the Beef" can cause a crash on PC, it might happen because you have companions following you in the same room or because you looked or talked to him before 4pm. [verified]