I believe it was after I had forced a re-download of the entire update directory and turned my graphics settings very low.
My notes aren't great on the variety of settings that I've tried, but I'll record a few particularly relevant points and follow up if there's more that seems useful. I've been trying to eliminate variables, but of course there are quite a few to address, and it takes a while to re-launch the game and run the mission. It does not write a crash log (as it has in other less-predictable cases), and the debugging logs I know about haven't been written to for minutes (or more) before the freeze. Sometimes a loading spinner starts, sometimes the background crowd noise starts, sometimes it stays completely blank but regardless the game stops responding and has to be forcibly terminated. Upon entering the elevator, the screen goes dark for the transition as expected.
In 'The Contract' storyline of GTA Online, during the finale mission of the 'Nightlife Leak' chapter, the character must take an elevator from the Casino Parking Garage to the Penthouse Floor. (Hopefully describing it here will at least be more helpful than Reddit. I'm still not certain whether this problem is specific to WINE, but the limited data I've been able to gather suggests that it might be.