I havent done the exact swap you are talking about, however I did install an 83 300 SD motor and trans into a 85 500 SEL.

Yes it can be done, takes lots of time, but not to much money.

Since i had both cars beside each other, i was able to install everything I needed from one to the other, Including ignition switch, Driveshaft, and flex discs, Trans crossmember etc.

In the W126 chasis everything mounts up, the electronics for the trans are the same, you will need the cooling system from the diesel as it has provisions for the oil cooler.

As I said, i have done it in the W126 chasis, and im currently swapping a 300 SD engine into a 84 500 SEC, and im planning to swap in a 83 300 SD motor into an 83 380 SEC, late in feb.

On my SEC im not swapping gas tanks, but I did on the 500 SEL, the tank is probably the same, the only reason to swap it is because of the gummed up fuel that the diesel will disolve.

The trans shifter will work fine, as will the throttle, you will probably need to remove the fuel pump (if there is one)..

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1984 Euro 300SDC, (4spd standard)
1986 Toyota Landcruiser Diesel HJ60 5spd X2

Gone but not forgotten (some sold, some stripped)
1983 300 SD, 1985 300 SD, 1983 240D, 1986 300 SDL, 1985 300 SDL, 1983 300 D, 1984 300 D, 1985 300SD, 1987 300 SDL, 1983 300 SD, 1985 300 TD Euro, 1983 380SEC, 1990 300 D, 1987 300D, 1982 300D, 1982 300D, 1994 E420, 1987 300 TD, 1987 300 D, 1984 300 D