Why can't the stock suspension handle the tires?
The stock suspension was calibrated using many factors but one was the grip of the Michelin PS2 or Michelin Sport Cup (special formulated with less grip). Most street tires will still work within the suspensions optimal range. The stickier the tire the more you are starting to operate outside that optimal range. At some point the cornering forces are too great for the spring and shock/damper to control. They start to loose control of the movement, may fall behind in correction, may over heat the fluid and basically start to flail. Plus you start to get unwanted body roll. This can cause you to ride on the bump stops, as the car rolls the negative camber starts to become more and more positive = loose you contact patch. In addition I believe the suspension geometry is less than ideal at full compression.
So an ideal set up with the stock suspension would be a less grippy tire. I would buy front camber plates and run 275 in the front.(AD08 or RE-11)
BTW Coilovers do not necessarily have to be harsh on the street. They can be set up for your individual needs and calibrated to the specific tires you want to run. KW has an ipod adjustable set up.
For reference: I have a 2700lb. E46 M3 with full aero running Yokohama race slicks. It takes 1000lb. front, 900lb. rear springs with MCS triples to control the corning forces that are near 2 Gs. The E92 M3 stock suspension + Yoko slicks wouldn’t even come close to controlling the motion. Its just not designed to do so…..