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Hmm...just ordered my truck and didn't realize the 120 outlet required the truck to run, but I guess I should have known that!
Ok, it might work without the engine running so try it after you get your new truck. I've never tried it without running the engine.
Still, there are electrician friends here that can tell you how many watts you can burn off battery power for a specific amount of time using the inverter.
I moved the fuse in the engine compartment to make upfitter switches 5 and 6 into always available regardless of run state, then pulled 6 ga wire (enough for 45 amps) from switch 6 through the console and terminated in powerpole connections on the back of the console.
Added a Semlex true sine wave 300 watt inverter and a cable long enough to reach the bed. Light loads can run for a few hours with no issues on the truck batteries. Added some more power points as well, including one in the cross-bed toolbox, on the same circuit so stuff can charge with the truck turned off.