Your 2-way Fridge, your temperature controlpanel, and your interior lights ALL operate on your camper's 12-VDC battery. The battery is kept charged by your camper's Converter (Charger). A standard 30-Amp service is made up of two 11-VAC lines to provide the 220-VAC for your Air Conditioner.
So, my first question is; are you using a standard 15-Amp to 30-Amp adapter made for RV's? If not you could be not powering the side that drives your Converter.
Also, make sure your camper's 12-Volt Cut-Off switch is not in the wrong position.
As to your Inverter, it also uses the battery's 12-VDC to make 110-VAC for your TV, etcetera.