What's a dielectric cooler?
Igloo, Coleman and other manufacturers of these thermoelectric coolers will all tell you that they recommend 2-3 hours maximum running off a vehicle starter battery with the vehicle not running. Igloo, Coleman, one of them, makes a portable power pack for picnics and stuff that will power the cooler away from the vehicle, and it's a 17 amp hour unit for about 3 hours of cooling.
Most thermoelectric coolers are designed to cool as much as 40 degrees below ambient temperature. That's fine for when ambient temps are between 68-80 degrees F, which is also the recommended operating temperature for most units. But when the temps get above 80 degrees, or the humidity is above about 60%, the cooling capacity of these coolers drops dramatically.
At one time, way before expediting, I had an Igloo 40-quart thermoelectric cooler that actually had a thermostat on board, and it worked great. It even had a low-voltage cutoff which would cut the power to the cooler when battery voltage dropped below a suer-selectable voltage. You could use it as a cooler or a warmer by switching the orientation of the power plug, just as you can with many of the truck stop coolers, but the neatest thing is that the cooler would cool or heat, automatically, depending the setting of the thermostat, which prevented things from freezing when operating in cooler ambient temperatures. But they apparently no longer make those. The thermostat quit working after about 4 years and no replacement parts were available.
The only thermostatically controlled thermoelectric coolers that I know about are the ones from Koolatron. They're nice, digital thermostats, will heat and cool automatically to achieve desired temperature, but not cheap. The 52-quart P85-PC3 will run you about $390, while the same P85 without the thermostat will only cost you about $150.
Edit: As Leo found out, in the long run, a 2.5 amp EdgeStar becomes far more cost effective than a 4.5 amp thermoelectric cooler. Still, don't run an EdgeStar off a cranking battery for more than a 3 or 4 hours. Get an aux battery for it.