We see the bathroom door in the first scene after the opening credits, where Kirk chews out Spock for his behaviour.
However, nothing says it has to be a bathroom door. Maybe it's just a narrow closet?
I think the narrow closet seems to be the best bet.
As to where is the bathroom - we have the bedroom section:
1. either somewhere behind the red curtain or
2. on the door side where we cannot see. Chapel and Kirk don't appear to go towards that door for some reason and opt for the entrance/exit by the "narrow closet".