Unless you are plugged into shore power, your A/C unit is not going to come on. However, the thermostat for it works on 12V hince the "click". As you have found out, running things on the inverter can easily lead to overload or a low battery situation quickly. Use of high current appliances, like a microwave, can quickly cause exactly the situation you saw. Next time, either plug the coach in or start the generator.
Roger & Mary
2017 Winnebago Navion 24V
2014 Tiffin Phaeton 36GH (Sold)