First i want to say - that i have no solution to your problem. It seems like it is temperature related and that could mean there is something wrong with your coolant temperature or your fuel temperature sender (G81). As you have done a lot off replaceing parts you had to dig deeper to find this fault.
One thing i would recommend when dealing with a VAG car is a diagnostic program called VCDS as its so much better - than those simple OBD software you can get.
Your diagnostic boy should know this program.
To diagnose a problem car is a difficult task - even if you has all the right equipment you wouldn´t be able to diagnose the car, if you cant interpretate the result you get from different diagnostic test done on the car.
What you need to diagnose this problem is a good fundemental understanding of how the system works and operate. You have have found there is a problem with PD injectors by using the Picoscope and that is a first good step in the process of diagnosing this problem.
But how are the PD injectors controlled - what others inputs are needed for the PD injectors to inject the correct amount of fuel ?
All I´m trying to say is - that sometimes you has to think out off the box to get the full picture.
I know this isnt much help, but it is a start.