It is unlikely you have an inverter in that unit unless added by the previous owner. The converter uses 110V AC and produces 12V DC and unless you had shore power connected, it was doing nothing, even then it would charge the batteries, not discharge them. More likely, the trailer battery has a bad cell, low on water or something and it drained the truck batteries. If the electric brakes were on via the breakaway cable, you wouldn't have been able to tow the trailer.
One other possibility is the refer, depending on model, some have a switch hidden inside the door frame at the top. It controls a 12V DC heating element in the door gasket to reduce moisture condensation, and it works even if the refer is turned off. So it can drain the RV battery and then the truck battery if it is on.
Anyway, it is never a good idea to leave the umbilical cord hooked between truck and trailer for very long w/o the engine running or shore power connected.
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Larry Day
Texas Baptist Men-Retiree Builders member since '01
13 Silverado 3500HD D/A, 2wd CCSB srw, custom RKI bed
11 Cruiser CF32MK
https://www.picturetrail.com/dayle1
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