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This one could prove to be one that is not able to be remedied without actually being able to see it, but I will give it a try.
The fact that it will start back up tells me that is most likely not ignition related - but we need to keep that option open.
My first thought is that it may be suffering vapor lock.
Is the fuel line routed too close to the engine or exhaust manifold?
One thing you could try to rule out or confirm vapor lock is to install a fan that blows on the carb and fuel line. If the set will continue to run with a fan blowing on it, we have most likely found the problem.
Another possibility is a bad fuel pump.
To test it, let the genset get hot, and shut it down.
Immediately remove the fuel line from the carburetor and place it into an empty container.
Crank the engine for a few seconds and watch the fuel flow. This will not have a particularly strong stream, but it should be 'pulsing' as the engine cranks.
Does it appear to have enough fuel flow?
Another possibility is that the stepper motor controller is overheating and shutting the set down.
Here is how to test it:
1. When you click on the link below, a new web page called 4shared will open.
2. When 4shared opens, scroll about half way down the page until you see a BLUE ‘Download Now' button on the left side.
3. Click on the BLUE ‘Download Now' button. Do not click on any GREEN buttons.
4. Clicking on the BLUE button will start a 20 second timer.
5. When the timer expires, a BLUE link that says ‘Download Now' will appear directly above the timer.
6. Click on the blue ‘Download Now' link above the timer to start the download.
Click here to download the file.
If none of this finds the problem, let me know and we can check some other things.