Diesels need the diesel transmission. They're programmed to start in 1st gear (gassers of the era started in 2nd) and have different governor settings internally. Will the 560 tranny work on it? If the starter is on the driver's side like the OM603, probably, but it'll drive like crap.

Try servicing the transmission (drain pan and torque converter, change the filter) and replace the K1 accumulator spring while you're in there. The K1 accumulator spring heals a lot of wrongs with 2-3 shifts.

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Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]