My wife has an 03 Cooper S. It "overheats" randomly. I recently put a new thermostat in thinking that was the issue. I can have it run all day and be okay. I drove the thing aroung for 2 days with heater on, then ac on with no issues. I leave town and it overheats on her. The heater won't work and the temp will climb and push coolant out the overflow. The engine itself wasn't hot. I can shut the engine off and start it and it will be back to normal. Playing with the heater controls seems to set it off. I tried bleeding it and seem to have gotten all the air out. I was not able to access the thermostat bleed screw though.

 

Finish bleeding it....it has not been done...you never opened the thermostat bleed valve....
Then hope it works....
Similar sounds to a failing head gasket.... But overheating can also CAUSE a failed head gasket...

 

That was my first thought, blown head gasket. Probably need to do a compression test and coolant pressure test.