In addition to the continuity error about the anamorphic lens seen in the preview theatre two different versions of the rushes are shown in that scene. When Olivier and Vivien Leigh are watching they are looking at a very wide cinemascopic version but, when it cuts to a different angle as they stand up to leave, the image is more of a 16:9 ratio.
In the opening sequence, Marilyn appears singing "Heat Wave" in the screen of a movie theater. She sang that Irving Berlin song in There's No Business Like Show Business (1954) with a totally different set and costumes.
In the screening room sequences, a close up of the projector lens reveals an anamorphic lens, which would be projecting the image in Cinemascope. "The Prince & The Showgirl" is not a wide-screen movie and rather shot in full academy, 1.37:1 aspect ratio. Furthermore, when the dailies are shown on screen, the image looks closer to 1.85:1, much wider than the original format of the movie.