@advocatux Thanks. Yes, I was aware I have to add the "bluetooth" policy group to the .apparmor-file:

{
    "policy_groups": ["bluetooth"],
    "policy_version": 16.04
}

However, I probably got confused: It seems that the AppArmor-configuration only gets adopted when the version number in manifest.json is increased (I don't know if this bug report is related).

Anyway, with this .apparmor-file and after increasing the version in manifest.json, the error ("Cannot find a running Bluez") is gone.

@Flohack
I this case, adding

"template": "unconfined"

to the .apparmor-file did not seem to be necessary to get Bluetooth-access. Thanks for the hint about running apps as unconfined anyway.