Question about Apple Pencil (2nd generation)

The Apple Pencil gen 2 does not work with my iPad Air gen 5 from the start, although it should according to the compatibility information. The pencil is only recognised and connects when it is docked. However, it does not charge (always 64%, although it should be charging according to the display). If you remove the pen from the dock, you cannot connect it via Bluetooth and it does not work at all. I have shut down the iPad several times and tried to re-establish the Bluetooth connection several times without success. Has anyone already had this problem and been able to solve it?