I just replaced my Pirelli P6000s with a set of Bridgestone Potenzas on my W124. They were the Veloce Sport versions. I had been unhappy with those tires anyway, and felt they wore prematurely.
I had tried to get a shimmy corrected when I got the Pirellis...after four tries at getting them balanced, re-balanced, etc., I gave up and lived with the shimmy.
The shimmy disappeared once I switched to the Bridgestones.
I say...LOSE THE PIRELLIS!!!
2009 ML350 (84K) - Family vehicle
2001 CLK430 Cabriolet (71K) - Wife's car
2005 BMW 645CI (124K) - My daily driver
2012 Mustang V6 (60K) - Daughter's car