The bathroom does not need a doorway, as you correctly presume, and accordingly, the corridor leading into it does not require one either. It is considered “the corridor to the bathroom,” and it is not associated with the living room.[A]Agur B’Ohalecha 19:12-14, based on Chazon Ish.
However, if the corridor also leads into another room other than the bathroom, and that room requires a Mezuzah, so does the corridor. In addition, if one uses the corridor for storage, e.g., if he has there a towel closet, the corridor needs a Mezuzah in its own right. Nevertheless, the doorway to the bathroom is exempt.[B]Heard from author of Agur B’ohalecha. Compiled by Shmuel Kraines.