Sounds like something is getting hot and loosing its connection - old GM cars used to have a module in the distributor that would do this - but your mini does not have it.
You might look at the battery connections - in the trunk there are 2 positive (red) battery cables and 1 negitive cable. The pos. cables are used to run different things - the small cable runs everything but the engine / fuel . The large cable with the big red block on it starts the engine and supplies power to the fuel system ( I think ) -- The big red block is a explosive disconnect that will turn off the power to the engine and fuel systems in case of an accident . The small cable maintains power to the hazard lights / windows / etc...
So, if I had to guess - I would check your battery connections first and check that red block for power after the car warms up - It might be the problem.
PS - I just re-read your post - you have a cooper not an S - I don't think you have the double positive battery cables.
According to my bentley manual - you should check 2 things
1. The clutch switch - which you already did
2. The EWS ( drive away protection system ) - try a different key - or have the system checked by......you guessed it - your local MINI dealer....