You should have simple K-Jet (CIS)on the European 500SEL. Cold starting and running depend mostly on the control pressure regulator (otherwise called warm-up pressure regulator).
If you haven't done it within 20k I would replace your spark plugs. You should view them to tell the story on what's happening to your motor. Look for differences in color and carbon build-up. Set your timing at least 5 degrees above current setting (factory setting that is). These two things will probably give temporary benefit to starting. If it isn't running properly you will have reduced vacuum and as a result high modulator pressure and hard shifts. Fix the running before dealing with the tranny (except if you also should service it).
If you would like to know about testing the warm-up pressure regulator let me know. It will require fuel pressure gauges!
Bosch Master, ASE Master, L1
33 years MB technician