Hm, I'm not sure how you can send dbus messages between different users.
Did you check that your user has this, or that the mopidy service has this as well. For me, I have this set when logged in as a user, but it is not set for mopidy when running that as a systemd service.
I tried setting it for mopidy explicitly, but then I get a permission error:
Failed to connect to bus: org.freedesktop.DBus.Error.AccessDenied: Failed to connect to socket /run/user/1000/bus: Permission denied
I don't know if you can allow this in some way or do some kind of authentication.
But what I think should work is running mopidy as a user service instead of a system service. Then mopidy would be run as your own user, and should have permission.
I have thought about if we should include a user service, so it would be easier to run it this way. One problem with the user service is if multiple users are trying to use the service with the same port. But since it's useful for single-user machines, and you can configure the port to be other than the default, it should be much of a problem. We should still include the system service either way.