It's tough for me to compare the two. I haven't driven a 4 cylinder ti in 3 years. I will say the Cooper S off the line at an autocross is slow. I don't have the technique down but I believe you need to keep the revs up, then drop the clutch. It takes a little while for the Cooper to get up to speed. As they say, there is no replacement for displacement. I have no experience with a DASC either.
The two cars are very different and you can't compare them.
I love them both but I'd be heartbroken if I was to loose my ti.
I will say a MCS will pass me on the track in my 328ti.
I've been passed by a well driven 318ti too.
I think the Mini shines once it is up to speed, not off the line.
The Mini also handles better in rain and snow then my ti.