We get stuck on the 'end of a cord' a few places, and here is a method to get the AC working without blowing a fuse -
1) turn the FAN ONLY on to high.
2) after it is running for a few seconds, QUICKLY turn the switch to low AC.
This has allowed us to run AC off a 20 amp breaker, 15A outlet with a ~50 foot 12Ga cord at the house. Trying to start both the fan and compressor at once blew the breaker each time. Using this method, we had the AC running for a few days in the driveway before the trip.
We do have the 13.5K AC, not the 15K. The extra size may make or break this method...dunno.
Frige would run on propane with it, but not 110v with the Air Conditioner. It would run for a while, but we would the breaker tripped sometime later.
As far as an extension cord overheating, if you are not using the entire length, uncoil it to allow it to radiate heat. A loose bundle on the ground may cool OK, but I use the cord reels and have had a few 'wake up' calls when I saw how warm the cord got on there.
I now see that you are using your shore power cord with an adapter. So extension cord is HD enough, DEFINITELY check the adapter several times when you start using it. I have had a couple cheap ones that dont engage fully, and can roast the cord end.
I finally made up a proper 30A service for the house, but the above method would run the AC for a few days while Mrs V got the coach ready for the trip.