I had to step back to Bullseye, because I had to have the possibility today. I will also stay on Bullseye for the next time, because of this.
What I can say: I do now believe that this is not the problem of network-manager or network-manager-gnome because I installed network-manager now from Bullseye-backports and it's now the same version as in Bookworm and all things work as expected.
