Bob,
I don't have an inverter in my truck, but I do run a generator in the bed sometimes, and have a 120v cord slung over the tail gate. My factory charge line only provides about 8-10 amps to the trailer, barely enough to overcome parasitic loads.
I'm not worried at all; if the trailer disconnects from the truck I'll have bigger problems then a 120v cord being ripped apart.
Besides, my generator and your inverter both have circuit breakers on the outputs, catastrophic damage to the cord would trip those breakers.
Regarding your question, I'm not sure about adding the second battery... Use the battery you have, and put the isolator between it and the inverter, such that when the engine is off, the inverter is unpowered.