You could power the TC with your truck batteries. However, truck batteries aren't designed to be discharged and brought back up. Also, you would potentially be stranded if you discharged too far.
Add at least one battery for the TC (you'll need it later any way), change the bulbs inside to LED and add a way to charge the house battery. You'll need a disconnect so that the TC won't continue to drain the truck batteries when the truck isn't running. You could potentially charge it with the truck, but better would be to add a solar panel or bring a small generator.