My issues with sluggish behaviour are almost gone: I used the exact PID values for Tilt Rate und Tilt Angle from this video:
viewtopic.php?f=32&t=508
Much better now. With these values as a starting point I will continue tuning ..
So it seems to be a PID issue after all.
(The video suggest reducing Tilt Angle D but this alone did not solve my problem, I also had to tinker with other values and finally decided to simply steal the settings from the vid)