Now that you put it into more relative terms, I have to say I have a similar problem.
My car already has the M3 rear sub-frame bushings, all linkages except the rear lower control arm are already M3 arms and I have a limited slip. But under hard acceleration in a corner with good traction I can feel the rear "squirm" and the car steers in a "funny" way. It doesn't quite cause me to leave my lane, but it certainly makes my car wiggle.
It does feel like something is loose/broken or otherwise allowed to move. Just can't find what it might be.
I do autocross and haven't had any problems during an autocross. But it might just because the corners are different during an autocross and I don't have much time to "pay attention" to the problem.
So if you do find a solution, let me know