Thinking through it, if you are using the 115v plug I would guess you have to find a way to keep the inverter powered. My guess is that power to the inverter is turned off when the truck is off. Find a way to power the inverter directly to the battery and the problem is solved. WARNING - wiring the inverter directly to the battery could cause you to lose the ability to start (if the battery was drained to low).
My fridge runs off of both 12v and 115v, so I just plug it into a 12v outlet. I've never had a problem with the fridge turning off. There are settings on my fridges for low/high power consumption, so a battery won't be drained.