The heat pump is the a/c, and the a/c is the heat pump. It's the same compressor running in a different mode, moving heat in or out of the RV. So if one mode works, the other should too. Of course, there are electrical relays involved to change the direction in which the heat is being transferred, but you say it works ok on shore power? The a/c doesn't know generator from shore power - it's just 120v electric.
May I suggest a very controlled test to be sure of the symptoms? You should be able to get the a/c running on shore power, then set the t-stat high enough so it cycles off. Then start the generator, which will override shore power. Turn the t-stat back down several degrees and see if the a/c comes on. If it does not, shift to heating mode and see if that mode comes on. If it does not, there is a power problem to the roof unit, e.g. no 120v output to the circuit breaker for the heat pump & a/c unit. If it does come on, we will have to analyze the generator power output with electrical meters (frequency, voltage and amps) to figure out what is different about it vs shore power. It maybe that voltage or frequency is off enough that some electrical device simply isn't responding. If you don't have access to this sort of equipment or have the skills to do it, it will be time for professional help.