I just brought my 2012 R58 Cooper S JCW to the dealer and demonstrated the thump when shifting from 1st to 2nd quickly, like you're racing. It sounded like a motor mount or torque arm broken or weak. They checked everything and could not find anything and could not duplicate it again.

Any thoughts what it might be?

 

I agree with your assessment. I had an Eagle Talon that would thump on a high torque shift (1-2 is a quick boost shift). It was a loose motor mount after a clutch job. Sitting in the shop they couldn't find anything. A quick run around the block and i demonstrated it. After telling them what I thought, we looked it over together and I found the loose nuts on one if the mounts.