No Run
I got left road side with symptoms very similar to what you describe about 4 years ago. Rail would charge, engine would fire and run for a few seconds then die. There is a relay bolted low on the brake booster assembly and kind of hard to see. This works in concert with the fuel pump relay located at the ECU. A PO had jumpered this and that wire had come loose (found it looking in the dark and moving wiring, saw the arc) I used a VW relay to replace and all has been well since. Probably something completely different, yet your symptoms are so similar....my post at the time...
"As me bride and myself were under way to sunday brunch, the coupe died.
(this was last winter) It was a fine sunny day. To make a long story shorter,
it then snowed heavy. Friends took us home, provided refuge for "Sienna"
I returned with tools and a tarp. She no wanna start no way as I had no fuel pump actuation. Had to get a tow back to my shop, after the snow melted a bit, to do proper diagnostics (that means that I failed in the field) With superman x-ray vision, I found that one relay that had been jumpered,
(previous whoever)and the jumper had come loose. Naturally just loose enough to make the deal "intermittent" This relay is hard to find, in the engine bay, as it is mounted to the inboard side of the brake booster, low, lot's 'o stuff in the way. Well needless to say that the relay was toast (hence the jumper) Bosch #0 332 003 014 This part is "unobtainium" where I live.
Anyway, here was the fix...#0 332 003 021 a VW part readily available. But... you must gently rotate the center spade 90 degress (10/51) Voila'
021 becomes 014.
I would love to hear more stories. As the parts get fewer and far between ..."
At the time I said "intermittent" But as I now remember, she would fire and then die. The observation of a short start from the cold start squirt is valid. Sort of like juicing with starting fluid. Shade tree stuff. Start the engine with the air cleaner off, minor mist with starting fluid just enough to keep it running. Tells you to concentrate on fuel issues, with d-jet this means electrical components. Sounds to me you are lucky enough to have the parts substitution down. Good luck to you.