It appears you have eliminated all of the fuel related issues. That would leave the generatoor safety circuits as a possible reason for the unit shutting down.
The Low Oil Level Sensor is the major reason that generators shut down when your sure there is not a fuel supply problem. Make sure the unit is fully servied with OIL. If the problem persists disconnect ONE wire from the low oil level sensor and see if the problem persists. If you still have the problem inspect the black or green wire from the Start run switch all the way to the Air shroud on the engine looking for signs of chaffing.
If you can not find anything then you will have to re-focus your attention on the fuel system once again. If you have a solenoid on the base of the carb.. the solenoid can be WEAK and stop remaining open once hot. This will cause the engine to lean out and die.
You should also check fuel flow to the carb. The flow should fill a spary paint can lid in under 10 seconds to the point your worried about it overflowing. Most honda engines have a metal sediment bowl under the fuel selector. Remove the sediment bowl and dump the contents. Check fuel flow again.
- Carb. With the float bowl removed fuel selector ON check fuel flow again by holding a spray paint can lid under the float. It must fill up in about 10 seconds to the point your worried about over-filling the lid. If you do not have a good flow rate here the float needle float needle rubber seat and the fuel manifold are the prime suspects.
I know this generator has the Honda GX270 engine. Some have solenoids (Item 49 this link)
^- No details!
Honda engines by model and VIN can be found here:
At this time I think the low oil level sensor (sending uint) may be the culprit causing your problem.
Item 2 this link:
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