We have a fiver with a Samsung residential fridge. First of all, they are one of the most energy efficient ones out there. Will 2 6 volt batteries run a 1000 inverter all day? I seriously doubt it. The inverter requires a certain amount of battery voltage to run then it shuts down therefore shutting down the fridge.
We precool our fridge in 90 degree FL heat about 2 to 3 hours prior to travel, load it up and then go. We have traveled 9 hours with it running with no power. But of course, the truck is maintaining the battery level.
Keep in mind, too, the fridge will cycle on and off once cooled to the proper temp but the inverter is still running drawing from the batteries.
I recently installed a 160 watt solar panel to offset the inverter draw on the batteries while in storage and traveling in the event we have an extended daytime stop over.
As far as an overnight stop with no power, it may stay cold through the night if you keep the doors closed. Another reason I installed the solar is to help boost up the batteries, maintain them in storage, and in the event of a power failure in a campground.
Another option is to add another battery bank.
I have had the residential since August 2014 with no issues but we do not boon dock. We absolutely love it and would never have another RV fridge.
I know I have NOT answered your question in technical terms but I know enough to tell you that NO two 6 volt batteries cannot run the inverter/fridge all day. Hope this helps.