With respect, your question suggests you're not fully confident with electricity. For all you know the unit could have been dumped because it was shorting on the mains input. You really ought to get it checked by someone that knows what they are doing.
However, before you run it off your inverter, check the power consumption of the cooler (it will say 'X watts' somewhere on a label) and check your inverter output. Coolers aren't usually very high so, unless your inverter is small, it should be OK.
I believe the Outwell coolers run off a Peltier device so there's not a lot can go wrong with them (no pumps or motors etc.) so, if it's not working, it's probably the power input. If it works off 12v but not mains it will be the mains PSU unit; if it doesn't work of either it will be the DC power driver. The other areas to look at are the switch and the thermostat. Unfortunately, by the time you've paid someone to fix it, it will probably cost more than a new unit.
IMHO it's got to be worth a punt for the cable, if it is dead you can always sell the cable on. As Broch said there's not a lot to go wrong. If anything it could of been tossed simply because the owner was unhappy with the amount of battery juice it used.