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Suggest for the sake of your battery that you set the battery protection setting on 'Hi'. 11.4v is into damage territory. (Portable fridge manufacturers are notorious for setting protection levels far too low, ARB are by no means alone in this).
I'm not familiar with the Bosch C7 charger, but it appears to be a pretty fancy smart charger, & as such should be ok to leave connected to your crank batteries & switched on if you are leaving your fridge running. If it has a 'supply' mode, get the batteries fully charged & then switch to supply mode, if not just leave it in float mode. It will work out when to revert to bulk charging rates. If the latter just keep an eye on your crank batteries, assuming they are wet type batteries, & top them up with distilled water to ensure the plates remain covered. This is because they will be getting charged to 100% rather than to the usual 70% to 80% that alternator charging does. In so doing gassing & gradual fluid loss is normal.
I have a couple of ARB fridges, including a 60 litre one, & know that the digital temperature readout can be misleading. This is not just an ARB fridge thing though. If you fill the fridge as much as you can with bottles of water or similar you can be more confident of the readout. I've found that with an empty or partially empty fridge it appears to get down to temperature rather quickly. If you have as little unused space as possible the internal temperature will remain more constant with less cycling of the compressor. When planning to use mine I usually fill the fridge (or freezer) with frozen bottles of water from our household freezer for 24 hours before refilling with food from the fridge or freezer to ensure a more stable temperature. This is far more efficient than using the 12v fridge/freezer to do the initial cooling/freezing.
Cuppa
2006 4.2TDi ex-Telstra Remote area Camper. Patrol Sold after 11 years of ownership Replaced with 2006 OKA NT Expedition Truck. Cummins, Allison & lots of goodies A Nomadic Life (Blog)