#1, Don't slip the clutch, let off the gas! The clutch is not for traction control. I have no idea why you would slip the clutch in reaction to not hooking up. That is a horrible idea.
#2, Limited slip diff
#3, Sticky tires
There's no magic want for this. 350+whp is a lot to put down through two 245-275mm street tires in 1st gear. Don't peg the throttle so hard in 1st. Leave TC on until you know how to drive the car.
2009 BMW 135i 6MT Sport, AFE intake, Cobb AP, Apex 18x8.5+9.5, 255/275 PSS