I've researched this a lot since I've been looking to add solar/battery power to my RV.
Can you run a heat gun off an inverter, yes, you can run pretty much anything of an inverter, given you have a big enough inverter and a big enough battery bank. That being said it would be incredibly expensive and not very practical to do so. Running it off a car, no.
Just run the extension cord or get power from somewhere else. If you really want power there get a small generator like a 2000 watt one or something.
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