Honda eu3000 overload running carrier air v for 5 minutes

I guess I'll ask one more time and then give up.... As it appears you don't want to answer the question.

Is the generator actually dying after five minutes with the A/C? Or is it just stopping power output.

Your very first post said it overloaded and shut down. If the engine is actually dying then it is NOT from being overloaded as they don't shutdown when overloaded. This will steer your troubleshooting a different direction.