Any output less than engine off voltyage (12.45 in this case) means the current is leaving the battery rather than into it as should happen with an alternator. The ABS is sensitive to the alternator to establish a running engine. It will shut off at voltages below 12v.
The alternatot on your car should be able to handle all those loads at idle (not the case with cars before 1990). handling means current flow into the battery, which means voltages higher than engine off battery voltage.
As I pointed out the ABS controller processes the wheel speed signal. When it shuts off the signal output dies. I think you have a bad alternator or regulator/brushpack.
Bosch Master, ASE Master, L1
33 years MB technician