Air conditioners are the one thing I have seen that do not seem to come in a 12V DC version. There is no technical reason why they could not, but there doesn't seem to be much demand, so no supply.
I may have misjudged your intent. If you want that much power out of your inverter then you need something close to what I did in the Stealth Camper. I thought you were aiming at something lower cost.
It may be that you are not yet fully calibrated in terms of what constitutes simple living in the RV world, and how much it costs to live the same life style as in your stix and bricks (at least off grid). Simple living in the RV world is often no inverter, no air conditioner, minimal batteries, everything in the house runs off of 12V and if you need AC to make coffee you fire up a small portable generator like the EU2000i. The house batteries charge off the alternator with a relay or a diode, or they plug in a tiny "Battery Minder" to keep the battery charged when sitting empty in the driveway.
A 3,000W inverter and the batteries to support it is fairly high on the luxury scale for most RV's. Even expensive rigs often have much less because they don't plan to live off grid much. They basically drive from camp ground to camp ground. Your needs are mostly "on-grid" as well, but you can spend as much money as you want.
Residential window units are tempting because they are cheap, but they may not take the pounding as you drive down the road. A tiny little air conditioner like that will not cool you off very quickly either. It would have to run basically all day to keep things livable in hot conditions. If you have been traveling and stop, the house is likely to be very hot inside. It might take hours for a small AC to knock it down to something livable.
I hope all this comes across as friendly input, not bossy.