The 12v Amp draw on the cigarette receptacles is limited as pointed-out by D'Arcy. If the box is already cold, or filled with cold objects, then you have a different Amps/Hour. There are some higher amperage 12v plugs with 2 prongs. That would work close to what hard wiring it would provide.

You can measure the draw with a pretty inexpensive voltage meter too, Craig....

It's possible that you have a bad compressor or motor too. I think the polarity is obviously and issue. IT'S POSSIBLE THAT HE PLUG WAS REWIRED INCORRECTLY, not the wire to the receptacle (or vice versa).

I'd CHECK THAT BEFORE YOU start changing things. You may have things crossed-up a number of places.