You need to understand your batteries better and owning a new RV will certainly help teach you.
I’m not saying your batteries are not the problem but “12.5v or 12.6v” is almost fully charged. 12.7v is fully charged. When driving or when plugged in or when running the generator your batteries will be being charged at over 13v. It varies but let’s say 13.4v.
So you look at the voltage meter and see 13+ volts. When you turn off the charging (unplug, turn off gen, turn off engine) you still see 13+ volts on the meter. Hence, you think your batteries are fully charged. But what you’re seeing is just a surface charge.
So to know your actual state of charge you must let the batteries rest - no charging and no big drain - for an hour or more.
Once you do that then you’ll know the condition of your batteries and you’ll be better able to see if they are the problem or not.
If the are 12.1v or lower that’s the problem. To fully charge them up don’t expect a quick one hour recharge It will take 4-6 hours at least.
About that fridge, it will help if you start it before a trip while plugged in. Then when it’s cold load it and leave it plugged in. That way it will use less power to keep the food cold than to get the food cold when starting out.
Take a look at the YouTube channel “the Fit RV”. They have a Travoto and have done some videos on living with a Travato. Though they don’t have the same fridge as you.
2017 Winnebago Adventurer 37F
2016 Lincoln MKX Toad