how would I fix that? my inverter is 3000w. I thought that would be plenty.
IMHO, You are not going to get any satisfaction from trying to run your A/C on inverted power.
BUT, what is happening to you is that when the load from the AC is on it demands more voltage from the batts than they can provide. Even Though they are fully charged to 12.7 plus, they may only be able to supply 11.7 (example only). Your inverter may have a setting (that might be adjustable) that tells the inverter to shut down at 11.8 (example only). This is called the "Low Voltage Cutoff".
I experienced this problem with a 1500 watt microwave. I could not understand it. I had a 3000 watt inverter, with no reason to shut down. When I started to look into it, I found that my Low voltage cutoff was set at 12.4v. It does not take much to pull more than your batt system can deliver. I adjusted the inverter LVC through the interface to 11.7 and cured my problem. Even a temporary dip below the setting will disable the inverter.
The "Low Voltage Cutoff" is designed to disable the inverter in the event that the charge of the batts drops below X charge. But temporary loads can cause the inverter to sense the low voltage even though the batts remain completely charged.
Let me state the obvious, in case you are not applying the fact that 1 AC amp is 10 DC amps. 10 AC amps is 100 DC amps. What this means is that your inverter must consume at least 50 amps DC to create 5 amps DC. So it should be easy to see why your batteries may pull down when the a/c is activated.
Good luck