It goes to the trouble of showing you the battery voltage at the end of your last trip, so it would be rather rude of it to reduce this to useless before you got back to the car! I reckon VW would have thought of that one.
Mine never gave me any trouble, of this sort, while it was fitted and the car was left for a few days, sometimes, so I wouldn't consider battery discharge to be a potential problem.
You cannot connect with the device with the ignition off.