Currently I have Nitto NT05 305/30/20 installed on rear (343 M6) and am telling you the grip is perfect.
305 needs at least 10.5" wide rim while your 343 M5 rim is 10" wide on rear, but there are many people in this forum have installed 305 tires on 343 M5 and they claim it's fit so you should be fine...
315 R888 is another ( lets say better) option but then you need 10.5-11.5" rim