Culber is not the CMO.
We have not met the CMO, nor any medical officer that appears to be Dr. Culber's superior, but on one occasion Culber says the CMO needs his assistance.
Stamets: Aren't there actual people on this ship requiring your attention, doctor?
Culber: Actually, the CMO does need my help with an Andorian tonsillectomy.
It is possible the tonsillectomy was invented to escape the situation at Stamet's suggestion, but it doesn't make any sense for Culber to invent a CMO.
I was unable to find a video clip but I found a reference on Memory Alpha that also documents the exchange.
Stamets asks Culber if there is someone on Discovery who requires his attention and the doctor states that he is due to help the ship's chief medical officer with an Andorian tonsillectomy.
Incidentally, tonsillectomies, or references to them, are still common enough in the 23rd and 24th centuries to warrant their own Memory Alpha page.