I'm no expert, just a life time of experience.
If your compressor works on house wiring, then it's your generator or you may be you are trying to use a wimpy extension cord.
My 3500 watt RV genny has 2 110 volt outlets, but they are only rated at 15 amps each. However the RV outlet, when set on 110 volts, is rated at 30 amps. If I try to start a small motor on the 15 amp outlets, the genny doesn't like it nor does the motor.
So, I bougth an RV adaptor to convert it to 110 v plug, NO problem with genny or what ever I plug in to it.
Another possibilty is the air compressor. I had a noisy oil-less air compressor. For some reason it would sometimes trip s 20 amp breaker in the house. I had no love for that compressor. Gave it to my son. Compressor lost compression and it ended up at the recyclers. Try a different air compressor. I have a 1.5 hp portable, not oil-less, that will handle 2 nailers. Had it over 20 years. It has never blown a breaker in the house. Cap start motor.
I use my genny to power my well if I have a power failure. Above ground 3/4 hp pump. It took 200 MFD, ACV, to power factor it. I did that to assist the genny. If you had to buy 4-500 MFD, you may change your mind and buy a gas powered air compressor. My contractor has 3 gas air compressors.