Should we say, "O Allah, I seek refuge in You from all evil and evil-doers" upon entering a multi-purpose restroom, or is that specific only to relieving oneself?

The remembrance (dhikr) mentioned upon entering the lavatory is supererogatory (mustahabb) and not obligatory. It is supererogatory for everyone who enters the bathroom to recite this remembrance, whether they intend to relieve themselves or not, due to the presence of devils in that place. It is also supererogatory to say "Bismillah" (in the name of Allah) upon entering.