I don't see a huge difference between a trip to the car wash with engine and transmission in place and a trip to the carwash with engine and transmission on a trailer. There'll be a few places you can't access with the engine and tranny in the car but I'll bet if you take a couple of ramps and a creeper, you can get a huge percentage of grease off the engine and transmission with everything in the car.
I take mine and wash them off routinely in the car wash and they turn out pretty clean even without ramps and any attempt to get underneath.

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1977 300d 70k--sold 08
1985 300TD 185k+
1984 307d 126k--sold 8/03
1985 409d 65k--sold 06
1984 300SD 315k--daughter's car
1979 300SD 122k--sold 2/11
1999 Fuso FG Expedition Camper
1993 GMC Sierra 6.5 TD 4x4
1982 Bluebird Wanderlodge CAT 3208--Sold 2/13