First of all, a 30 amp outlet IS a 110 outlet. Actual, they are all 120 volts as that has been the standard for decades. For some reason the old 110 / 220 volt lingo won't die.
So no, the outlet choice didn't make the wire get hot. Are you running the AC thru the inverter's transfer switch? It's possible that due to all the voltage drop in the small wire feeding whatever outlet you were plugged into that the voltage at the inverter was below it's lower limit. Therefore it may not have actually connected to the shore power and was instead power your air conditioner from the battery. If that's what happened then plugging into a 30A RV outlet should cure the problem because of the theoretically larger wiring and thus less voltage drop.