AC doesn’t work after lithium/inverter install

I would look into compressor burnt first.
I would not try to run off an inverter. Why would you do this? it runs batteries down fast, and you will not get much AC time with batteries anyway... AC units are a big reason people own generators.

Your AC has nothing do do with your new battery system, so you can stop looking there (except you could have done damage using the inverter). Have you tried un-doing all the AC upgrades and trying it again?