@CowboyTaco The temperature stabilized after the first couple hours and is now within the +/-4. It is also empty so you're probably right.
On page 5 of the manual is says "Note: A battery charger may only be connected to the battery when the Fridge/Freeze has been disconnected from the DC source." Seems strange.
I assume they mean an AC charger shouldn’t be used while you are drawing power from battery via DC. You’re fine with your car though since it’s DC. I wouldn’t worry about it and my guess is it’s there for people in other countries who use this in their homes/AC to charge.
This is how I interpret that as well. Don't have the fridge running while you have a car battery charging from a wall charger. The battery being charged by the alternator in everyday fashion should be perfectly fine.