I have worked it out: When I pull the HDMI cable out of the monitor and immediately plug it back in (whilst the monitor is still switched on), in 1 out of five times the aspect ratio resets and the OSD becomes fully visible again.
It's not an ideal solution, but at least I know how i can work around the issue. And now you know as well
Albeit, this may be a fine work-around, I'm not completely happy: First i thought the issue had to to with the Black Pearl getting confused. However, since exactly the same problem exists when i connect the controller to my Sony HD TV, I now think that the DJI controller's HDMI sometimes gets confused about what to send to the HD display. To exclude other factors, I swapped the cable and tried again => Same problem appeared but luckily also same work-around worked. To rule out a hardware defect of my main controller, i tried with the second controller: and again: Same problem/same work-around. Conclusion: I'm 99% sure that is a minor firmware bug in the light-bridge part of the controller.
I spoke to DJI Hot-line about it. The Hot-Line did not have an explanation and insisted that I "debug my settings until i get the best video results". I don't see how this will resolve a HDMI bug in the controller, but choose not to go into lengthy debate with the friendly hot-line employee who at least tried to help me, but sadly lacked the knowledge/experience how to deal with a bug.