My assumption is that the compressor runs on DC, so the AC side input through a transformer and then is rectified to DC. This means that if it works on AC it should work on DC as well, which suggests an "upstream" problem somewhere between the plug and the circuit board or a low voltage issue.
Do you have a multimeter? If so, I'd check for continuity along the DC input path (with the fridge unplugged from all power sources) and see if there are any loose or broken connections.
That said, it's suspicious that the fan comes on for a few seconds and then turns off. Does the digital temp gauge light up and then turn off as well? It is entirely possible that you are hitting the built in low voltage protection. I have an Edgestar FP430 and have run into this a few times.
What are you plugging the DC cord into? These fridges can draw a reasonable amount of amps on initial startup, and if the fridge if fed by small wires there can be enough voltage drop to trigger the low voltage protection. The low voltage cutout is adjustable, so as a quick test you could turn it all the way down and see if the fridge keeps running.