If there was no issue with battery depletion before the towbar was fitted then it logically follows that it must be something to do with the towbar installation, but what?
What puzzles me - being a VW owner myself - is why the installer didn't use the VW kit. I've had two Passat Estates fitted Westfalia towbars with the VW wiring kit and, including reprogramming of the ECU, they charged me £540 inc VAT in 2015 and £600 inc VAT in 2018 - as against somewhere around a grand at a dealer. The disadvantage of not using the proper kit is that it doesn't communicate with the ECU so said box doesn't know you are towing. With the correct kit it will detect when something is hitched up and change the display on the Infotainment unit, it will activate the Trailer Stability Control system, and as Damien says it will show graphically on your LCD dashboard any light failures. For the record on most VWs if the ECU has been reprogrammed there will be a change in how the warning lights on the dash operate: normally the Traction Control light and ABS light go out simultaneously, but if a caravan is attached and plugged in one of the lights stays on for about 2 second longer than the other. Also engaging reverse gear which usually shows a graphic of the car and the parking sensors on the Infotainment unit will instead show a car with an A-frame on the rear of it.