One of the key efficiency factors was being able to pull and hold a vacuum. And obviously the more efficient the engine that pulls the vacuum the better.
If it's a modern high efficiency unit, it probably already pulls a vacuum. If not, it may be worth physically hacking.
That being said since the system doesn't use batteries to store energy it might make sense to use any leftover energy to pump the air out of the fridge even if you don't reach a full vacuum.
Or maybe just build up the vacuum externally throughout the day until a certain time, then use it to pull the air out of the fridge for the night when it won't be opened and won't have power.
But you do make a really good point. In cruising, people who choose to have a fridge have systems for this.
It is usually to get saltwater really really cold or frozen in some kind of containers: it is popular to use PVC pipe, capped off of course, filled with seawater (or a mix of fresh and sea).
You store your food in the freezer and charge up these pipes, cycling them on to the top of your food in your cooler (not powered) once every 3-4 days.
This: "1 kilowatt of solar panels has managed to cool the fridge down close to standard fridge temperatures."
Is kind of silly to me though... because that's an "incredible" amount of power.
Pessimistically that means 3kWh/day, which if you're storing it efficiently (in a battery or as thermal mass like OP) gives you a nominal 125W supply