Most of us have 15amp circuits at home. I do and I got a 10/3 100ft extension. 10/3 will prevent any meaningful voltage drop on a 15-20 amp circuit over the 100 feet. It is this drop (which heats the cord) that will affect your RV voltage.
I may be wrong, but most air conditioners have cutoffs that will shut the compressor down if the voltage is too low (actually if the compressor starts to overheat due to the low voltage) and will reset after cooling off. I was unaware that any modern A/C would get permanently damaged by low voltage. Consider brownouts and high usage power conditions when line voltage can drop below 108.
My A/C runs cheerfully on my household circuit. Originally, I had a 12/3 100ft cord, but it bothered me that there would be appreciable voltage drop, so I got the much more expensive 10/3. Nothing seems different, but I have more peace of mind.