Recommend you do not start replacing components like the FSR, but rather do a proper test for parasitic draw. It isn't the shift light, and if it was the FSR you would hear the blower fan running after you turned the key off. The radio is supposed to work without the key (it is timed). It is possibly something that you don't even use, like the telephone module. A parasitic draw test is the only way to find it.

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue