Thanks Vin for your suggestion. I think I found the root cause in the meantime:
I tried lower values for the pot, and also Arv's suggestion to replace the pot with a 2K2 resistor to ground (this works very well btw.). Still massive contrast difference on key down. Filtering the RF with a parallel capacitor did not help, as there was still a DC offset.
I decided to mount the LCD away from the PCB, as I suspected it to pick up RF from the PA stage just below. Immediately, the problem went away. When I mounted the LCD back on, you guess it, the problem still was gone!
I checked the space between PCB and LCD, and noticed that a flap of the LCD frame was cutting into the windings of L2:
I assume this was the original cause for the problem. I bent the flap away some more to stay clear of L2.
Now I can enjoy my qcx even more! Thanks to all for your help and suggestions.
vy73, Peter HB9TVK