A bit late to this thread, but this is my experience using a fridge powered from the cigar socket in the boot space. I have the Auxiliary battery fitted with associated C-Tek from INEOS.
I drove to Munich from home (UK - circa 900 miles) so I think I can assume my starter battery and auxiliary battery were fully charged. I have a small Dometic CX35 in the rear, plugged into the cigar socket. I had no other electrical appliances connected.
I left my car in a hotel car park (underground - so no heat from sun) and after 40 hours the fridge had already gone into “safe” mode to prevent discharging the batteries too much. The fridge temperature had already risen to 11 degrees (c) so I guess it had already been off for an hour or two.
Car started as normal with no indication of problems (no warning lights etc). Sadly, I didn’t think to check the state of charge on the central screen, I was more pre-occupied as to whether the car would start or not, and was so happy when it did.
Since leaving Munich last Wednesday and driving back in the direction of home, I have used the car every day and the fridge has operated as normal.
So my conclusion is that the car needs to be used daily to be certain the fridge will continue to work.
Going into this, I had expected that I would have been sharing the load across two batteries and therefore be able to run the fridge for longer without driving the car. Clearly this was wrong.
Hope this helps the debate