Actually, there's three fuel pumps on our 2005 3.0d E53.... One electric pump in the tank, one in-line electric pump in front of the tank and then the main high pressure mechanical pump bolted to the engine (HPFP).
Here's the in-line pump...
This "injector" fault could be caused by any of them. Or a clogged fuel filter. Or a failed fuel rail pressure sensor, fuel rail pressure regulator or fuel flow regulator.
A scan of the DDE error codes with INPA or PASoft etc. would go a long way to reducing the number of possiblities...