After half an hour of confusion and tinkering, I learned that my problems were due to switching from a Qt 4.8 build to Qt 5.3, and not clearing the build directory completely.
macdeployqt does not work properly if there is an app built with a different Qt version.
In the midst of wondering what is going on, I also ran into a number of display mode switch issues. It looks like I’ll finally have to drop the deprecated OS X display mode change API calls and just use the OS X fullscreen mode. This means the display mode is not configurable via Doomsday (which is what Apple recommends, I suppose). However, there might be a way to kludge around the glitches (black screen remains after mode switch) by adding some further delays and other tricks… Doesn’t seem like the right thing to do, though.