We have a 2011 Countryman and the Vista frame is a tight fit even in that, even with the wheels off. It might be even more of an issue for the standard Mini. Take the frame to a Mini Dealer and you'll be able to try the various permutations to check what works.
We never used the bassinet 'mobile' and just had the car seat (Besafe iZiGo Isofix) on the frame until very recently where we now use the pushchair attachment. The frame/pushchair just fit the Countryman. You will not get both in the boot of the Cooper.
You may have to 'lose' access to a seat if you need to go anywhere once you little one is a bit older. Our little man is now 9 months old so we leave the car seat in place and use the frame+pushchair attachment. That doesn't leave a lot of space...all our long or shopping runs are now done in the X6.
ISOFIX is standard on all cars now and has been for many years. By 2011 it will be on the car by default.
Current: 2011 X6 40d & 2012 Countryman SD All4
Project: 2005 E46 330Cd M Sport convertible