1800 watt Inverter drained battery while driving?

Check battery cables connections first. What is truck voltmeter readings?

 

even slightly loose battery connections will kill your batteries.

 

Did you check alternator, if it’s working your batteries should not go dead.

 

If it is 15 volts it’s overcharging the batteries.
Should be 13.8 - 14.2 volts.
The fridge is not enough of a load to be the problem.
If you recently wired up the inverter then I would suspect loose wiring connections as mentioned.
First time I wired one to truck master shut off switch. A few weeks later the truck died going over a bumpy railway crossing at highway speed while -20 outside. Fortunately I checked the last thing I did, one of the nuts on the shutoff switch was loose.

 

If it’s running at 15v, and can’t keep up something should be hot.
Look at the batteries, are any swollen, do you smell rotten eggs. A bad battery can suck all the power.

 

Sounds like a bad battery(s). Both high heat and severe cold are hard on batteries. If you're in Indio, heat could be hurting them. I live in Yuma Arizona, I have plenty of experience with heat causing battery issues.