Running AC off of 12v means LOTS of amps (100+ —>200 depending on other minimal loads). It *can* be done, but LA voltage droop will be significant, Lithiums less so. Regardless, 100s of amps isn’t great for lots of things, not the least of which is inverter circuitry. Few minutes for microwave, or hair dryer, etc isn’t a concern IMHO but I wouldn’t recommend it constantly for AC in high heat. Some do successfully; but my recommendation would be to set up a 24v system, or better yet 48v, if you want to run AC off of batteries a lot.
I have 540AHr lithium and a 3kW whole-house boost inverter, not saying I’ve never run AC off of my batteries but I have a 4500 Westinghouse generator that I use if I want to run AC. Even that only put out ~20A+ when we were camping at 9300ft a few weeks ago.
Back to OP original issue, I also have MicroAire’s on both 15kW ACs. Doesn’t change current draw while running, but eliminates the inrush current and loud bang when they start up, significantly reduces the voltage droop from batteries OR generator which are not as close to ideal sources as shore 120v.