Headlights on to operate van fridge?

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When towing (2014 Honda CR-V) I discovered that my fridge was not charging (Thetford fridge showing fault code 6 - no or insufficient 12v supply). The towbar fitter found that it would only charge the fridge if the headlights were on while towing - presumably putting lights on is raising the demand for voltage which then feeds through to fridge. The car has a dedicated loom and a separate charge line extension kit from Right Connections, I believe. The fitter is trying to find out more info but I am still awaiting a call. Any suggestions would be very welcome. By the way the rear parking sensors are not automatically disabled either. Apart from that...