I wish I had a good answer for you,I guess it could be possible the FSU is defective or not making a good connection, sometimes the pins on a connector can bend out of wack.The car should go to sleep when its turned off,have you added any after market components or altered the wiring in any way? These cars are so packed with electrical components its really hard to find the problems.
Maybe try pulling up the wiring diagram and see just what else is on the same circuit, maybe something will jump right out at you. look at the REST feature, its the only thing I can think of that works with the key off other than the alarm.
Sorry I don't have a solid answer for you, keep us all posted on what you find cause it really helps all of us understand our Bimmers better. Good Luck!!!