Is the prayer for entering the toilet said only upon entering the bathing area, or does it also include entering the facilities?

It appears there is no difference between entering the lavatory (bayt al-khala') and the public bath (hammam) regarding the desirability of saying "Bismillah" (in the name of Allah) upon entering them. This is because the purpose is to seek refuge with Allah from Satan, and this is not exclusive to the lavatory. This is supported by the hadith narrated by Tirmidhi: "To conceal what is between the eyes of the jinn and the private parts of the children of Adam when one of them enters the lavatory is to say 'Bismillah'." As for the remembrances (adhkar) related to relieving oneself, it appears they are specific to the lavatory.