Here is a manual I found on the Onan website.
I am not familiar with the 4000 but have owned a 5500 and a 7000.
Both have electronic voltage regulators that control the frequency and voltage. If either get out of spec either from load or a mechanical issue it will shut the generator off to protect it.
I am pretty sure yours has this too.
I am pretty sure you can get fault codes from it too that will probably tell you why it shut down.
Happy reading.
You could also take it to a service center and demonstrate it.
Sounds like you are overloading it to me. With the ac on I would not try to run anything except lights and one other thing.
