I say RCM. Engine Control Relay in particular. THis one could have a bad ECR Relay - so resoldering the board would not fix it.

But the battery drain part has me wondering. Only if a relay on the board was stuck closed would a RCM cause a battery drain - there just isn't anything on the board to be a current sink. All the switched power and grounds to throw the relays come from the PTCM or the ignition switch itself.