I fully agree that some fridges do need the actual supply lead upgraded ..yes waecos were the real offender for this.
However in this case I don't see it actually making much difference if any at all.
The lead will have a pin configuration into the fridge and unless you are going to remake all of this then simply chopping the lead after the plug and joining it will not help in fact i would argue just introduces another failure point at the connection.
The weaco leads suffered from a poorly designed thermal fuse arrangement and as such the chopping of the lead had more to do with getting rid of this than the cable size, many ppl simply cut the that end off and shortened the same cable with a positive result.
So Id say leave your fridge lead alone.. inspect the ciggy plug connection for any movement and if you wish to and I advise too at a later date simply cut the ciggy plug off and replace with a better plug Anderson etc.
In most cases it is the actual plug as opposed to the cable that is the problem. the cable from the end of your supplied fridge lead to the battery is a whole another issue.
baby it through keeping your voltages as high as you can ..constant small charges are better than infrequent long ones so just run that big 4 wheeled generator every 2-3 hrs for a bit, turn the fridge up when the car is running and turn it down when its not running will also help the battery out.
Best of luck mate
Jet