@Bluegayle
I loaded up my old save file so I could test it out.
My Sim had a movie gig Of Tea & Treachery. Just like your picture above, the private dressing room was available to him for the movie gig. I quit the game and reloaded the save. This time, I canceled the movie gig and got a commercial gig, Salty Suds. I fast forwarded until it was time to join him and this is what the set looked like:
https://i.imgur.com/lN0PqlL.png
Despite him being level 10 in the acting career and being a global superstar, there was no private dressing room for the commercial gig. So, I do think it's tied to the type of gig and not career or celebrity level.
I didn't test it out for the daytime and primetime television gigs. I can try to test those out, too.
EDIT: I just tested it out with a primetime television gig: Dignity & Decorum. The private dressing room was available.
EDIT 2: The daytime television gig Wild Flames did not have the private dressing room.
So it seems like if your Sim is doing a higher paid gig like primetime TV or movies, they will have a private dressing room. Lower paid gigs like commercials and daytime TV will not have a dressing room.