The air conditioner needs a lot of power. At a minimum, the circuit you are using to power the trailer needs to be rated at 20 Amps, and that may not be enough if the converter is also charging the trailer's battery and it definitely won't be enough if you are trying to run the refrigerator or hot water heater on AC power too.
However, I'm surprised the air conditioner shuts off before you pop a circuit breaker in the house circuitry. I would think it would overload the house circuit. I think the extension cord you are using is problematic too - even the heavy duty ones I've seen are rated for no more than 15 Amps. Possibly, the air conditioner has a low current/voltage sensor that shuts it down if it "sees" it is not getting enough power.
Does the extension cord get hot? Does it get hot at the plug ends? If so, it is being overloaded. A hot cord will have increased resistance too, that will reduce the power available to the air conditioner as it heats up.
Hi-Lo 1707T - Tire Minder TPMS on Tow Vehicle and Trailer, 300W Solar Battery Charger, Equal-i-zer WDH
2014 F150 Platinum 4x4 3.5L EcoBoost SCrew