Yep, it's the low voltage zap causing the inverter to shut off. It might be a weak starter battery, but in some systems an inverter will sense too low of a voltage regardless of the battery's state.
In any case, an auxiliary battery for inverter use is highly recommended. Connected via a battery isolator, the alternator will charge both batteries and you can drain the inverter battery without fear of not being able to start the truck.
Be careful of the isolator you get, however. Make sure it is a true isolator, and not a simple solenoid switch (a continuous duty starter solenoid). A battery isolator will totally isolate both batteries from each other at all times, with the only common "connection" between them being the alternator.
A solenoid switch works by "isolating" the batteries from each other, but only when the ignition key it turned OFF. When the key is turned ON, either to start the truck or while the truck is running, both batteries are connected, and all things electric will draw from both batteries, as well as both batteries will be charged by the alternator. But as long as the key is turned OFF you can drain your inverter battery and all is right with the world, right? Not really.
The problem comes when you go to start the truck, and you turn the key. The drained battery and the good starter battery are instantly connected, and the starter battery takes a huge zap from the drained battery in desperate need of power. The discharged battery will always drain power from the charged battery. It's usually not enough of a hit to prevent the truck from starting, but it will be with a weak starter battery. And, repeated zaps like that will lessen the life of the starter battery. It can turn a 5 year battery into a 2 or 3 year battery, for example. And the sudden drain can damage batteries, heat up wires and connections, even to the point of causing a fire.
So while a solenoid switch will work, the best bet, by far, is a true isolator. You also want one that uses diodes to let each battery (or battery bank) determine how much power it needs from the shared alternator.
Best isolators I know of are the "Sure Power" line of battery isolators. They are the most common, and can be found at RV dealers, auto parts places like NAPA, and the JC Whitney catalog. They come in several amp ratings. Easy to install yourself, or it's a half hour deal to have a service technician install it (an hour if you have a Bosch alternator, probably).
