Accessory mode or Inverter for 12V cooler?

Whatever the inverter is hooked to stays on. Maybe run a jack from those 12v terminals, keeping in mind tbat if you leave anything turned on without the truck running for too long you may have a dead battery to deal with.

Or buy one of those jumpstart boxes at harbor freight and power the cooler with that while sleeping, then change it from the inverter while driving.
If you're a company driver that is less likely to lead to trouble with the company.

 

The reserve capacity of those batteries should handle the load for ten hours, this is a six amp load with all the overhead. Three batteries, or four ... if there is an under voltage alarm, it starts with the batteries,

 

That sounds more like either the alternator is weak or you have mostly bad batteries.
You should have both tested to see where the actual problem is.
Or the voltage regulator...

The inverter uses the batteries, but if they can not be charged by the alternator while the engine is running they stay pretty well dead. If the regulator is bad it will not allow the charge to be sent to the batteries.

 

Yup. I only ever use heavy load devices (microwave, frying pan, kettle) with the engine running at fast idle. Never have a problem.
The one time I did have a problem I had a weak alternator. The diodes burned out on one side of the windings. During the day it worked fine, but at night it couldn’t keep up with the headlights on.