A quick search on google gives a power consumption of around 60 watts (which is 5A at 12v) so I wouldn't expect them to run for too long on a 12 volt leisure battery which are usually rated between 80-100 Ah*.
Assuming a 70-80% efficiency then you will be lucky to get 12 hours before the battery is dud and I would suspect it wouldn't even last that long. You will also kill the battery if you deep discharge it that much on a regular basis.
Using a 12v-250v inverter would just give even more losses as the fridge would just convert back to 12V so it would be worse.
Best thing if you have no hookup would be to take the coolbox with you if you go out in the car and the engine will keep the car battery topped up while you drive.
Of course you could just take less and buy as you need. A stream (or a bucket of water changed regularly) makes a good fridge
*In theory 80A for 1 hour or 1A for 80 hours but you get nothing close to that



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