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.
"Yes. The White Glove Society strictly forbids eating humans. But we weren't always the White Glove Society."
PCXbox 360 Targeting Chauncey during the quest "Beyond the Beef" can cause a crash on PC. And he can die in the pool of the hotel.
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.