Get a used differential and install it. I think that Brian was right. I have installed two difs and am in the process of installing another. One dif change was due to noise as you are describing, another to put a 2.47 in a 300SD and this one to eliminate excess backlash. The job is not difficult. It does not need any expensive special tools, but does require some physical strength to remove and replace the actual pumpkin. The process is outlined very clearly in the Factory Service Manual.

Good luck,
Tom

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1993 300TE
1984 300D
1984 300SD
1976 BMW R90S