There are several things you need to understand about your trailer's power systems.
1- Your camper's lights, alarms, temperature control panel, and other accessories run on your 12-VDC from your camper battery which is kept charged by your external campsite 110-VAC power to your Converter.
2- Your fridge relies on gas or 12-VDC to operate but the icemaker runs on 110VAC.
3- Your 110-VAC outlets operate on your external campsite power. And you have several receptacles that are on a GFCI circuit that can kick out at times.
SO, If you want everything to work you need to have that 110-VAC campsite power hooked up.
And, you need to check your camper's battery; 1- does it have water in it, and 2- is it charged up?
Your Air Conditioner does run on 110-VAC, but if the temperature control panel doesn't have adequate 12-VDC to it, it will not allow the AIR to run.
Start with understanding these things, and you can get your camper accessories operating properly.