Was watching a video on a 2001 E39 (manual 4.4 V8) which has a similar problem with the alternator light. The video features a few takes at Fluid Motor Union in Chicago and the owner of FMU (a very capable wrencher who has an e53 off-roader) opines it might well be the oil level sensor or the wiring to the OLS. IIRC he mentions that the OLS interfaces with the CAN-bus.

On the V8 the wiring for the OLS runs from the passenger-side wiring box atop the engine along the front of the engine all the way down to below the sump. That is 3-4 feet of wire in an awkward location and easily damaged. I'm not at alll sure that the wiring location on the M57 engine is the same but, since the electronics on the e53 and e39 are close cousins, it might be worth checking?

__________________
2003 4.4i black ext./black int./black headliner (kid's runabout)
2014 535d X-drive, M package, silver/black interior

Sold but not forgotten:
2009 E70 35d, black ext./black int./black headliner (sold 2021)
2006 4.8iS Le Mans blue/cream int./black headliner, SOLD in 2012 sadly...

Other hardware:
2015 Cayenne S (wife's new DD and definitely wroooommmmm)
2016 Tundra 1794 edition crew cab, luxo hauler
2005 Tundra crew cab, weekend hauler. Sold after 150k very happy miles.