Do Inverters powered through the cig lighter work for appliances?

Saw a bunch of plug in stuff at Pilot today. Hot water, skillet, slow cooker.

 

I run a crock pot off of a cig lighter inverter.
It powers it sufficiently while the engine is on.
If I attempt it while OFF.. it chirps in protest at the limited power draw.

Mine is a 400w Schumacher from Wal-Mart (though I doubt it's really 400w bc the watt/amperage thru a cig lighter can't convert to that level)
It's prob just a 240 in reality.

No way in Hades could it power a fridge, cooker, or any combo above the single pull.

 

We have a refrigerator that plugs into a lighter outlet. Works well.

 

Is that a fridge with a compressor, or just a cooler with a fan?
I tried a few over the years.
Had a 1000 watt direct wired, but the fridge would burn out its start circuits within a month.
Moved up to an 1,800 watt. That worked much better. But if I started the the truck when the fridge happened to kick in, the start overload would blow on the fridge.
Now have a 2,500 watt pure sine wave. Fridge has been going for two years, no problem at all.

 

I tried some of them 12v stuff on 2 FL's. After a few uses. I couldn't even get enough current to charge my cell phone. Ended up stuck with an invertor just to get 120 volts to charge.