Am not really in ABS systems connected with late model LIKEs.
First I would check voltage at my battery during running motor. Should be not more than 14-14.5 VDC.
More during a longer spell might shoot your battery.
Then, if it's at battery still more than 15V, I would change the regulator/rectifier as a first measure.
Also all connection should be checked for a irregular "ground" somewhere.