I really don't think the LS would ever pass a Cooper S. If you take the given (underated) horsepower rating on the Cooper S, it has 163. The LS only has 164, but weighs a couple hundred pounds more.
For more empirical evidence on the underrated engine in the Cooper S, look at Grassroots Motorsports experiences with a dyno. http://www.grmotorsports.com/minicooper.html
If you take 155 hp at the wheels and algebraically solve for 15% drivetrain loss, the car is really making about 182 horsepower at the crank, which is where car manufacturers tout their horsepower figures.
Similarly, their Cooper made 109 hp at the wheel, assuming 15% drivetrain loss, the engine is really making about 128 horsepower.
I would guess that a MINI Cooper S with a power to weight ratio of 14.4 lbs per hp would have little trouble passing a LS, with a power to weight ratio of 16.4 lbs per hp. (I used 2,700 lbs as the weight for a LS, this may be incorrect).