Where are the field formatter settings?
4k4's comment was helpful in that it indicated the correct behavior.  This isn't what was happening; the POST requests were returning correctly, but I assume they're not returning a correct part of the dialog to replace.
However, with this knowledge in mind, I was able to use Configuration Management to export the View and the Entity view display, pull the appropriate setting names from the Entity view display, and manually apply them to the View YAML and reimport.  It's imperfect, but knowing that this should be possible was a great help in finding a workaround.