Hi,
so my headphones are ***paired and working again***.
Thanks to Edelgard at the OEZ Apple Store, my AirPods Pro are working again. I am not sure whether this is the solution but here is what happened. My headphone case was checked and tested with another pair of AirPods Pro. Ultimately, the decision was: the pairing button is broken. However, when I put my own headphones back in again ***the LED started blinking white again***. I was then able to pair the case with my iPhone again.
So, if you have the same problem and another pair of AirPods Pro at and, give it a try!