Just went throuh the same thing, had power to connector and even heard the pump run for a second when I did the relay bypass, after that the car would start then cut out. I removed the pump and it also ran on a battery while not pumping fuel but I think failing under load so today I put a new pump in and the car started and ran well so it seems the pump can still run but not develop preasure. After the noise when I first jumped the relay the pump could not be heard.
1995 525i 225,000km Still stock for now