I think there are two separate problems.
The first problem is that admin-web can only perform modifications on the file-level, while the trigger running postconf -e is only available on the key-level and thus never run.
Reloading the postfix service does not have any effect either because there is currently no way to update the postfix configuration. This can hopefully be solved by providing an appropriate file-level trigger, but I still have to test this approach.
The second problem is that the AAPI service might not have sufficient privileges to call systemd. This is an issue specific to Debian due to the ancient polkit version that does not support JavaScript rules.
I'm not sure whether pystemd can solve this issue as the current systemctl exec implicitely uses D-Bus as well.