You didn't have to go through all that. You could have just told me the model number.
Fortunatly for you, the newer BGE's are my absolute favorites, and are very easy to work on.
Lets start with determining which subsystem is responsible for the engine not wanting to stat when hot.
Remove the air filter.
Start the engine, and let it run for a while, preferably with a load on it, to let it get good and hot.
Shut the unit down.
See if it will restart. Remember, we have to catch it while it is not starting, so if it does restart, ket it run some more.
Once you have it to where it will not restart, check to see if your are getting fire to both plugs. If not, we know we have an ignition system issue.
If you are getting spark, while craning the engine, start spraying in some carb cleaner. Do NOT use starting fluid.
Will the engine start on carb cleaner? If so, you have a fuel issue.
Let me know what you find.