Why do I need such humongous cables to the battery when a small cord takes the AC output fine?

Power is volts times amps (Watts = V x A). So if you have a lot of voltage you don't need many amps. Roughly you need 12 times as much current from the 12 volt battery as you need from the 110 volt AC outlet. Current is what causes cables to heat up, not voltage. That is why they use thousands of volts in power transmission grids.
The thing to do when you have lots of current is to lower the resistance of the cable. The larger the wire the lower the resistance. Think of the cable as a water pipe. A big pipe (wire) can carry more water (current or amperage) with less pressure (voltage), and will present less pressure (voltage) drop from one end of the pipe to the other.
Another consideration is how far the cable has to run from the battery to the inverter. Long cable runs are expensive, either in copper or efficiency, or both.