https://www.carobdcode.com/p0604-bmw...lf-check-fault
What I read is that the ECU failed a self check. Read the linked page.
I would remove the ECM and clean the contacts and replace, make sure you are holding 12v + trying to start, maybe have a jumper cable on the car when you attempt starting.
If the voltage drops enough during a start attempt the ECU will not send Siti nor fuel.
The fuel pump running key off is a new one to me. Are you sure it's the fuel pump?