Too much voltage drop can kill the AC, even with the 30A shore power cord plugged right into a house receptacle without an extra extension cord.
I just found out the other day that even with our TT plugged directly into a receptacle in our garage, our built-in vacuum cleaner kept tripping our EMS unit. The voltage was down to 98 volts! I was shocked to see that. The receptacle is at the end of a run of #14 ga. running up, down and all around the garage . (Good voltage using our dedicated 30A recept. in the carport tho.). So don't just assume that the voltage at the recept. in your house is perfectly fine and the issue will be just only with the extension cord and don't assume that going to a 30A extension cord will be okay either.
I would try a load test and measure the voltage inside your camper with a voltmeter. Just use an electric heater - 1500 watts would be good. Or even two of them. Keep in mind though that the current draw from a purely resistive device like a heater has an output that varies with the square of the voltage. For example, if you plugged 2x1500 watt heaters in and got 105 volts, you be getting 105/120 squared (rated output is normally at 120V) times 3,000 watts = 2340 watts and current would be 2340/105 = 22.3 amps.
Besides the full load running current of an AC unit, the momentary starting current is several times the full load current. If you do a load test and the voltage is low, but near 105V, the AC may not be able to start. If it looks questionable, you could consider a hard start capacitor. I read a Coleman manual once that said damage can occur at 105 volts and you don't want to be running near that, especially if nothing else inside is on like the MW, coffee maker, toaster, etc.
An Progressive Industries EMS unit would be a good investment to protect your camper against low voltage conditions which can happen anywhere - at home or out camping, especially during the summer with lots of people running AC units.
