I was also looking in to air conditioning when shore power wasn't available. I found that it requires a lot of power. I decided to stick to using A/C when I have shore power.
You can check the power requirements by looking at the amps the air conditioner draws. Amps times 120v = watts. Inverters are typically rated for peak watts, how much they can supply for a short burst. A 2000w inverter may only supply 1000w of continuous power. You'd need a big inverter. Take your watts, divide by 12v to get amps you will pull from the battery. This will be lower than real life. The inverter will eat some power. Take the amps times how many hours you want to run it and you'll have your amp-hours or ah. That will let you know the battery bank size.
I started looking at this for when I'm driving. I'm not this good of a diesel mechanic yet tho
Cat | Air Conditioning Kits | Caterpillar