The 12v plug that the fridge itself uses might just suck. On my edgestar, it used to shut down because of low voltage occasionally untill i zip tied the plug into the socket really tight. I think a proper solution would be to use a more secure plug and socket for the fridges. I think they draw enough to heat up the cheap 12v plugs since the plugs themselves are lossy.
My 12v outlets have like 4 ga wire going to them in the back and it still did what you are describing until I zip tied that plug tight.