I think we are assuming the generator is running properly and delivering the power it should at full load. Unfortunately many generators have valves that are out of adjustment which limit the HP of the engine to something below what it was able to do when it was new. This in turn will cause the generator to fall short on lock-rotor and full load amps. You would be surprised how many generators can’t deliver what they are rated for. Additionally, the power module may not be delivering the needed voltage when the AC starts simply because something like an AVR or throttle governor is out of adjustment. I would load test the generator to insure it is actually performing as it should and if all that checks out then start looking at the AC unit.