I've had this issue on my E39 previously where the battery connector wasn't fully seated. Visually it looked fine but it wasn't fully tight. There was always enough connection to run the lights and radio and everything just fine. But intermittently the car didn't have enough amps to start. Other times it was just fine.
But the E39 was also a simpler machine.
All signs point to a battery issue. Have it load tested and possibly reseated if it tests fine.
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