It may be somewhat impractical but you could use Star Diagnosis to monitor and record the voltage on circuit 15. It is not dissimilar to having an oscilloscope.

You would connect to the engine management controller, enter the engineering menus, find circuit 15 voltage among the "Actual values" menu items, and chart the voltage. The moment your engine quit you would then glance at the screen...

 

I will certainly check as a first test that circuit 15 is present when trying to start the engine with the key.

 

Well, I tested Circuit 15 and 15R and they are correctly switched on by the key as you turn it to positions 1 and 2 and are not disturbed by turning it further to the start position, 3. And still the engine starts sometimes but usually not. No idea where to go next!!