Is it permissible for the husband's mother to enter the bathroom in the wife's room without her permission, especially if the wife is present in the room and the bathroom door faces the room's door?

If the bathroom is private to you, or exposes your 'awrah (private parts), or if custom dictates seeking permission to enter it, then your husband's mother is not permitted to enter it without your permission. This is based on the Almighty's saying: "O you who have believed, do not enter houses other than your own houses until you ascertain familiarity and greet their inhabitants." (Quran 24:27). You should inform her of this gently, or your husband can speak to her. If there is no harm in her entering, then overlooking it is better.