Assuming that your unit is manufactured by Dometic, and is relatively new, 4000 W should be plenty of power to run one 15,000 btu AC unit. At least that is what Dometic says:
I little investigation might help us narrow down your problem. What happens when you go to Med. or High speeds when running the generators? Do you trip a breaker? Is the breaker on one or more of the generators or is it in the trailer? Does anything else happen?
Are you able to get up to Med. and High speeds when connected to shore power?
Are the eco-throttle switches on or off during your testing?
Are you confident that your cord isn't damaged, and that any "dongles/dogbones/adapters" that you might have connected to it aren't showing signs of overheating?