When this shuts down and does not restart, I assume that it will just crank. If this is the case, can you pull a plug out and connect it back to the ignition system then hold it against the engine block and crank the engine. We need to see if there is spark. I suspect you do not have spark.
If you don't have spark then we can go down that path. We have to figure out why the spark is going away when it gets hot. I suspect you have an issue with the Ignition Control Unit. These are hall type sensors and they are sensitive to heat so this is where we need to start assuming we don't have spark.
Let me know that.
Saturday, September 7th, 2019 AT 9:02 AM