I did another test flight, but unfortunately the issue with the C/L menu persists. I think I did a proper preparation, so maybe it is a bug?
There is one post from someone with the same problem on inibuild's discord, however no solution posted their either.
I already took does steps yesterday. Fresh install and clearing WASM folder. Unfortunately it didn't resolve the issue.
I'm going to check A330 Driver's tutorial videos again. Maybe I missed a step during the cockpit prep. It is obviously working correctly in his videos although I don't know which build he was using during the recording.