From: Rabbi Moshe Sofer
To: Rabbi Betzalel
The Chasam Sofer sides with the Bach's conclusion that a mumar l'avodah zarah retains a distinct halachic status from a non-Jewish idolater for shechitah purposes, supporting this from Rashi, the Rashba, and the Beis Yosef, while conceding that the Shach's critique of the Bach's specific Tosfos-based argument was valid. He introduces a novel distinction between bris milah and bris Torah to explain why such an apostate retains some Jewish status, and resolves a contradiction between two statements of R. Yochanan regarding efshar l'sochto by attributing them to different Amoraim.
Source: Shu"t Chasam Sofer YD §1