I'm sorry, I don't really see a solution...

The tyres are excellent and already quite wide, and you have a LSD...
By Kleeman it is a Detroit-locker, right? Not a LSD, but a differential locker...but that's the same effect.

Be gentle with the throttle in the first 2 or 3 gears...is the only solution I see.
You also can let the ESP full-on and enjoy a tenth of the power...

I don't think you can retro-fit a Launch Control, do you? And it is not really what you want...

You could try to see the technical data of a 65 AMG, with 620hp like your car, and see what are the differences with yours in the drivetrain/tyres, maybe you could find an idea here. Or have a look at some big Brabus to see if they have something you don't. Have you asked Kleeman if they have something for you?