Inverter issue's

That's it then. It's drawing over a 100 amps DC when you run the A/C. Probably have to get a generator to run it for a long time.

 

Dude, you are draining your batteries fast with that load, get a generator.

 

You don't have enough battery to keep up with what the ac needs.

You need a generator.

you double the amount of batteries you can double the amount of time before it starts beeping at you. That won't work for long.

You only have some much money in the bank for the wife to spend. It won't last long unless you are making money. You really need to be putting money in the bank to keep up.

 

First of all it's a square wave inverter.
Your AC unit isn't designed to run on a square wave. The compressor motor is probably getting really hot. I have a pure sine inverter That's rated at half the capacity of yours but it's four times the size, weighs 40 pounds, and is 5 times the cost. Something like that would probably be much more suitable for running an AC unit without shortening the life of the compressor.
If you're looking for 3000w inverter, this would be your best bang for the buck and it's made in USA and it won't shorten the life of your compressor.
https://www.amazon.com/SENSATA-DIMENSIONS-3000N-FREQUENCY-INVERTER/dp/B00FG5MARU
SENSATA-DIMENSIONS-3000N-FREQUENCY-INVERTER

 

Harbor freight generator and stand up AC unit installation.

Here is my solution. If you have room behind the cab get a bigger , cheaper gen then what I have. Also mine only works 7.5 hours. I went harbor freight , I can return it anywhere in the country. Extended warranty is $80.

If your frame rails are long enough put a side box on it for the generator.

 

Anything you plug in that has a motor in it, or in this case a compressor, needs 2 to 2-1/2 times it's rated operating current to startup.
Your inverter is probably large enough, but you don't have enough reserve capacity in the batteries to run it very long.
The fact that it works fine with your engine running proves you need more batteries.
The RV industry sells a Soft Start kit for air conditioners to help lower the load on the batteries when the air conditioner kicks in.