Dave I'm thinking out loud here, because I work on these from time-to-time, but I've never had to chase a choke issue, nor have I really studied it's operation....because they always seem to work.
That said, are you sure you have a problem? As long as you can manually close the choke door using that metal hoop that you pull out, the generator will start, and then my experience has been; push the metal rod back in, and the engine will let go of the choke door as it warms up, thus opening the choke door fully after a few minutes of run time. It sounds like that's what yours does.
So when it's time for a restart, you just manually close the choke again, give one pull, and the engine is running again.
That said, are you sure you have a problem? As long as you can manually close the choke door using that metal hoop that you pull out, the generator will start, and then my experience has been; push the metal rod back in, and the engine will let go of the choke door as it warms up, thus opening the choke door fully after a few minutes of run time. It sounds like that's what yours does.
So when it's time for a restart, you just manually close the choke again, give one pull, and the engine is running again.