La habitación tiene un cuarto de privado.

yo quiero la habitación individual con un cuarto de privado.

The right word that applies in the blanks is:

- La habitación tiene un cuarto de baño privado.

- Yo quiero la habitación individual con un cuarto de baño privado.

Explanation:

The translation of the sentences is:

- The room has a private bathroom.

- I want the single room with a private bathroom.

The sentences allude to a place within a room that could be private and, if you think about it, a room is a very small room to have something private, it regularly has a bed and a closet which in most cases belong a single person, which could be called "private", but if there is something that not all rooms have and that could be called "private," is a bathroom, which in a house of five members, regularly has only the main room, that is to say for the parents, while the children share a general bathroom outside their room.