It really seems like you have multiple copies of the file installed. Open up file explorer, go to your CFS3 folder and type the filename in the search box. See how many copies of 109e_prop.dds you have, and check to see if any of them have an all black alpha layer (that would make them invisible.)
What's odd about the bdp symptom you describe is that the game rebuilds the bdp files before it does anything to show you objects in the game. It seems like if it worked the first time it would work every time. It must be rebuilding the bdp files again on the second start-up, and that's when it rediscovering the erroneous version of the dds file again.