Start the car for how long? Starting drains more from a battery than a 15 minute run will replace. ALternators need to be run at about 1500rpm or better to charge adequately so sitting at idle at 700-900rpm will do little.
Add a charge line from the tow vehicle that goes to the battery and it will stay charged while being towed, even if the key is on. Had this same problem last summer as a charge line was not installed. got to 1st stop dead battery. Next morning we installed a direct charge line and had no further issues in next 4000 miles. I am sure you have a brake system of some sort dont you? When he siad they bypassed the cars elec system he meant they wired a harness to the tow plug that goes to the rear lights, same thing I have.