"Have you tried just replacing the rear diff gear oil with a good quality synthetic?"

The 124 4Matic and the ASD differentials are true limited slip differentials, and MB recommends their 85W-90 hypoid gear oil A001 989 17 03 10 formulated for LSD.

Change the diff fluid to that and run it for a few hundred miles or kilometers before replacing the differential.

Replacing the diff doesn't look like an easy DIY job to me, but someone who has actually done it at home will hopefully say something about it.