5,000btu AC unit will not run o 1,500 watt inverter. Why?

You first said #12 wire (very small) and above say #2 but that would not have fine wires so I think you missed the 1 and it's really #12. That's too small.
Check the voltage at the battery with a multi-meter at the moment the A/C tries to engage. I suspect it drops to less than 12V or you will find the tiny wires getting very warm trying to get that much juice through those wires.