I am puzzled why you might need a thicker cable for a replacement compressor fridge. One of the main reasons to change to a compressor fridge is the lower power consumption. I am about to make this change. My existing absorption fridge is a 170W model whereas its replace to compressor fridge is 80W. Perhaps it is to cope with initial start up surge currents?
This "puzzlelation" is precisely why so many compressor fridges have been installed with inadequate cabling which stops them working with anything but a fully charged battery and why so many owners end up starting their engines to start their fridges

. I lost count how many folk I spoke to at VW Festivals when I had a Trade Stand asking me why their fridge wasn't working properly in the van either they converted or had converted for them.
The issue is nothing to do with the running power of the compressor fridge (which is not 80W but more like 50W typically in fact) but as you have considered, the surge power/current is the issue for undersized cables, where the voltage drop on surge on thin cable can be dramatic and so much that the fridge detects a low voltage situation and refuses to start.
A couple of screen captures showing the difference between the surge current and the running current once compressor has started
IN-RUSH

And this doesn't actually capture the full power spike, which can be upto 20x the running power and is why if you are fitting a 240V Compressor fridge into a camper for off-grid use, you don't get a tiny inverter delivering 100W max, but need one that can provide 800W or more momentarily.
Post IN-RUSH

This was for a 240V Fridge powered by a Victron Inverter and even with good sized cables and a short run, there was still a voltage drop of >0.5V. Check on a on-line calculator such as
here what the voltage drop would be on say a 2.5mm2 cable on a run of say 3 metres and a current of 40A
This video shows the full power-on sequence for the Compressor Fridge. And note this is a sequence that is repeated many times a day.
0:05 - Inverter Switched On
0:20 - Fridge Plugged in
0:30 - Fridge Door Opened (so Light comes on)
0:41 - Fridge Temp Dial turned to Max - Compressor In-rush Current comes on
0:45 - In-Rush Current ends
Note that the power use you are seeing from
0:45 is the maximum power draw all the time the fridge is actively cooling, which is a small percentage of the time on a Compressor Fridge (this is what makes them so efficient in use).