That your cord is warm worries me a bit. Is it possbile something else is on, adding to your load? Battery charger(s), freeze protection heaters, a pump left on, water heater? A typical RV fridge draws on the order of 600W or so starting up, and then closer to 200 (Dometic).
FWIW, a true 14ga copper extension cord with 1000W load on it will drop the voltage about 3.5%, about 4 Volts. You probably want to be using a true 12ga copper extension cord, and yes, not in a coil.
Where is your GFCI? Is it an outlet, or is it in your cord? GFCIs do go bad.
All the best,
Peter