Not cooling means the compressor isn't running, but needing the genset instead of shore power points to other (electrical) problems. Need to tackle those separately, I think. Does the compressor start and run when using the genset? But still no cooling? What happenes when using shore power instead of genset? Does other 120v stuff work on shore power?
One possibility for the compressor not starting is a failed hard start capacitor. Easy and not expensive to replace. If you hear the compressor "grunt" and try to start but never actually get going, the capacitor is a good bet.