I believe if you get the right shim rings, it will work.
First, my experience on this: I bought what was supposed to be a W123 differential, low mileage, with guarantee etc. It fit the chassis (my '79 300TD) but then I saw my tires were wearing on the outer edges. The car appeared to be higher in the back. After a close look we discovered the diff was from a 300SD. After getting thinner shims (MB kit $50) the distance was properly compensated. The W123 differential casing was just a little thinner measured from side to side at the faces where side seals are. I couldn't believe the shim rings were that critical, but they are.
So, the reverse might be your problem and if you didn't take this into account. your axles might be too short where they meet the hubs, because I believe the 2.88 W123 case is thinner than the SD case.
BTW after putting a W123 2.88 differential in my wife's '83 300D it became a real pleasure to drive especially on the highway over 75 - 80 MPH, the engine really loves it!
As a side benefit, the speedometer that previously read fast became almost perfectly calibrated at no extra cost!
DDH

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'95 E320 Wagon my favorite road car. '99 E300D wolf in sheeps body, '87 300D Sportline suspension, '79 300TD w/ 617.952 engine at 367,750 and counting!