I would skip all the M3 bits and go right for better dampers/springs. TC Kline might be within range.
Consider Michelin Pilot Super Sports. A tad expensive but they really are "super" in every way (except maybe in the snow
). Hankook RS3 sucks in non-dry conditions. PSS is also quieter and lasts longer.
2011 BMW E92 335i, 6MT, M Sport, Le Mans blue - Michelin PSS, Quaife LSD, JRZ RS Pro, JRZ camber plates, Eibach ERS, full M3 susp, custom toe arms, Powergrid end-links, brake ducting, SS brake lines, UUC SSK, ETS FMIC, ER CP, Perf. Exhaust, work by VAC Motorsports