This does sound like a conundrum, and I have had the same symptoms before with my 1982 240D. I believe either air is leaking into the injection system when it is hot and depressurized, so that to get it running you have to purge the air, or the starter is getting ready to give out and at elevated temperatures the insulation resistance is such that you do not get the necessary rotating speed to fire the engine. There may also be some other modes of failure for the starter when it gets hot.
The first idea, air leaking into the fuel system when it depressurizes, does not explain why the air is not there when the car is cold. It is possible that it shrinks in volume and is small enough when cold to be flushed through the system without getting it airbound. Sounds like a reach to me.
The weak starter theory sounds better, and on my 240D I changed the starter and the engine mounts when I noticed the driver's side mount was broken in half. The starter was lifting the engine and kind of tossing it around the engine compartment, which was also making the actual torque available to turn the engine less.
I bought the "heavy duty" starter from FastLane, which is listed for the 300D Turbo, after confirming it would fit, and installed it with some new engine mounts. No more problems. Starts immediately hot or cold. Jim
1986 Euro 190E 2.3-16 (291,000 miles),
1998 E300D TurboDiesel, 231,000 miles -purchased with 45,000,
1988 300E 5-speed 252,000 miles,
1983 240D 4-speed, purchased w/136,000, now with 222,000 miles.
2009 ML320CDI Bluetec, 89,000 miles
1971 220D (250,000 miles plus, sold to father-in-law),
1975 240D (245,000 miles - died of body rot),
1991 350SD (176,560 miles, weakest Benz I have owned),
1999 C230 Sport (45,400 miles),
1982 240D (321,000 miles, put to sleep)