[Alpine runner]

Trying to help the dealer solve a problem for my mom. I know the combined wisdom here can be greater than one tech.
1) Car died idling in a driveway, I gave her a jump and it went to the dealer
2) My thought was alternator killing battery (because it died while running)
3) Dealer tested alternator twice, said it was good, replaced battery
4) car died again 2 days later while driving. SES light came on, no engine power, she thought there was still battery power. It started no problem when flatbed came.
5) Dealer said it was fuel pump, replaced under good-will.
6) Car died again while driving. Started up 10 mins later.

From what I've searched, I see "Faulty Intelligent Battery Sensor", and TSB for battery connection behind fuse panel, although that seems to pertain to 2007 MY.
Are there any other possibilities?

[E46driver]

I just experienced the same things your mom did except for it dying while idling.

I replaced the spark plugs and ignition coils and haven't been able to duplicate the issue.

Also, I don't know if your mom can give a testimony to this but if the car runs worse as the engine* gets closer to its normal operational temperature, her issue is probably the same as mine.

My car:
2009 328i n51
1044xx miles(was on original spark plugs and ignition coils)