I believe some have plugged the connectors into the wrong things before...maybe you did...nothing personal.
So, maybe therm connector is plugged into the exhaust solenoid...or cps. I believe that can be done.
Does sound like battery...a jump would answer that question.
Upcoming NYC dates: Jan 8 and 27. See Facebook for other shows and PM me if you can make one...love to see you there!
Mangofy your E46!