Salle de bains

Elle, 

In standard French usage it is always ‘ salle de bains “ with ‘ bains ‘ au pluriel. 

See Larousse link, under difficultés : https://www.larousse.fr/dictionnaires/francais/salle/70707#difficulte

and Robert

 https://dictionnaire.lerobert.com/definition/salle

A bathtub is not ‘ un bain ‘; a bathtub is ‘ une baignoire ‘.

‘ Bains ‘ refers to the ‘acts’ of taking a bath in this context. ‘ Salle de bains ‘ is effectively the room of/for having baths (washes).

In the sentence noted,  ‘ salle ‘ is in the singular with singular adjectival agreement - une salle (de bains) privative. As you expected, this represents ‘ a private bathroom ‘.

Bain(s) can have other meanings, depending on context. See links below for previous discussion and more information 

https://www.larousse.fr/dictionnaires/anglais-francais/bath/564781

https://www.wordreference.com/fren/bain 

https://www.wordreference.com/enfr/bathtub 

 https://kwiziq.learnfrenchwithalexa.com/questions/view/why-la-salle-de-bains-and-not-la-salle-de-bain-1

  https://kwiziq.learnfrenchwithalexa.com/questions/view/why-la-salle-de-bains-and-not-la-salle-de-bain-1

https://kwiziq.learnfrenchwithalexa.com/questions/view/why-la-salle-de-bains-and-not-la-salle-de-bain-1