Will the Fan in this A/C start and run when you select Low Fan or High Fan? NOT calling for cold air, just the FAN?
If so, while you're on the roof, checking and cleaning the A/C, open up the little panel covering the Capacitors. Should be at least two. One for FAN Run, Brown Wires on it. Labeled around 7.5-mfd on the side. And one for Compressor Run, labeled around 40-mfd. There is hopefully a third one, black, around 100-mfd. If there's the third one it's an optional Start Assist, as MEX mentioned. If Start or Compressor Run.
Compressor must have RUN, and it must be good, or you can see the circuit overload you describe. If it has Start Assist, that one also has to be good, but you can disconnect it (TAPE the Wire) for a test using shore power or the bigger generator.
If it won't start on shore power (true 30-amp circuit) on Run Cap only, carefully inspect RUN. If it looks swollen or its little relief plug looks disturbed, remove it and go to an appliance parts shop. "I need one of These." For $10-$20 you'll have the only likely piece to solve the problem. The other would be a new compressor. At that point you're better with a new unit.
Work on Capacitors with POWER OFF, ideally unplugged. Then short the two terminals together with a screwdriver before touching them. It's possible for one to stay charged with the unit stopped. They are meant to store electricity.
Please investigate and report back.
