I am not sure if the FSR, with the related blower fan fuse removed, will drain the battery...But easy enough to check....charge the battery up and pull the FSR...

If not do a parasitic drain test...There are like 6-8 things that can drain a battery on an E53..

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'99 E39 528i 5-speed 130k
'06 325Cic auto 115k
'05 X5 3.0 auto 173k-SOLD
'03 530i - manual swapped - 430k miles - Parting