My 2002 would leave two black strips before I put a LSD in it. So did my 2007 E90 335 before I put an LSD in it. I have driven pre- and post- 03/'08 335s. There are trivial differences with full DTC/DSC and partial DSC/DTC (quick button push). Long button push on pre- results in electronic nannies completely off. Post- keeps some braking when asymmetric wheel rotation is detected -- you can never turn DTC completely off in a post-. Get over it; there is no mechanical or electronic anti-slip mechanism in the post- differential. The final drive unit is exactly the same. "e-differential" is a minor software change in the DTC/DSC system. It steals the name of the real thing, an electronic limited slip differential that some Ferraris and Porsches have. You guys want to buy into the marketing BS, fine. Go racing with your "e-diff", especially off-road rallye racing.
Do most E9x drivers really need a LSD? Of course not. But you will find the M3 equipped with a real LSD.
To answer the Q regarding possibility of reprogramming, that is can of worms. I would guess that a new ECU might be part of the picture.