Update...I saw on peachparts there is a crank position sensor listed as a replacement part on our E300...I'd be looking closely at that since they do typically fail exactly as he described (under hot-soak conditions)!
http://catalog.peachparts.com/ShopByVehicle.epc?q=&yearid=1998@@1998&makeid=63@@MERCEDES+BENZ@@X&modelid=6384%3AMBC|1572%3AED|814%3AFT|6@@E300&keyword=sensor@@sensor&catid=All@@All&subcatid=P%3A242838@@Crank+Position+Sensor&applicationid=W0133-1716592&mode=PA
If that isn't it then I would replace the K40 relay. Even though he said it "looks OK" they are so frequently faulty and you certainly can't see every fault it is very likely to be the problem if the CPS is not.