OK - I'm going to post this knowing I'm going to catch hell. The last car I had with your symptom. Start when cold & will not start when hot suffered from low compression. When the car was hot everything expands & the engine wouldn't start. The engine does turn faster when push started. Other than that I would probably be on the starter not turning fast enough when engine is hot. Could it be the engine tolerances are so tight after it heats up that the starter is having to work harder? I know I've suggested 2 opposite views but I would want to consider all possible causes even if I didn't like them. If the engine is new & does have extremely tight tolerances the latter might be the cause. I have mainly drag raced most of my life & I have seen both instances with other peoples engines as well. I know you have been wrestling with this problem for a while & you have my sympathy. I know I've had problems myself that drove me nuts.
Dale
Dale