Inverter

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.

+1 inverter was trying to power a/c. I don't know what model or inverter you have, but the generic 2500 inverter doesn't list pass through power as an option meaning it doesn't have an internal transfer switch to turn off the inverter part of the box when 120V is available. Being rated at 20 amps output, the inverter may be capable to run the a/c. But as you've learned, the battery cables seem undersized.

Is that the wire to the inverter getting hot? Any chance the inverter is powering the air conditioner? And what size wire was getting hot?

Yes. Are the battery terminals bright shinny clean, tight and connector to cable the same same?

What is the charging amp capacity of the charger? Mine can charge 100A DC and draw 11A AC. Add a A/C to the mix and things can get hot in a big hurry on 20A.

When I'm on a 15A or 20A plug I throttle the charger down to 10% or maybe 20% and the A/Cs are OFF.

And 2x 220 Ah GCs are undersized for a 2500 W inverter.