What is the actual power drawn by the AC unit? Even if it were only half that of your inverter, it would still draw about 150 amps from a 12v battery. You will need a really big battery if you want to run it for very long. Even if you are running the engine at the same time, you are still drawing it down much faster than most alternators can charge it.
Please note, that the CCA (cold cranking amp) rating of a typical car battery is not it's amp-hour rating. While the CCA rating are often >500. the AH ratings are usually less than 80AH, some are less than 50. And, if you are drawing very high currents, the effective AH capacity goes way down.