I get what you’re saying about the concept of a mezzanine being part of the room (per 505.2), but I do not see a provision specifically granting an exception to rating elevator shaft enclosures that serve only a room and a mezzanine.
It seems odd to think of an elevator serving only one story (because the mezzanine is not counted as a story,) but the shaft is “connecting” less than four stories and would need to be rated per 713.4. Plus, as a typically unoccupied space, the elevator shaft would be a concealed space but can’t be fire blocked, which supports the need for the rated shaft enclosure.
712.1.10.2 specifically mentions vertical openings for elevators in parking garages, but 712.1.11 (regarding vertical openings between a mezzanine and the floor below) does not specifically mention elevators - if the intention was to include elevators I would think they would have mentioned elevators.