Trying to run an AC thru an inverter and powered by batteries and solar is a lesson in futility. Can it be done? Yes but 90% of the people who are familiar with the option will tell you to forget it.
I would suggest a pair of Honda 2000's. 70 lbs each and you can run one for just battery charging and normal routine power needs and add the second one to the system and power your AC too.
About $2000 for the setup.
If you aren't interested in running the AC at all, then solar again does become a popular option and its merits can be measured against a single Honda 2000.