Possible faulty cell breaking down in the battery?...........if it is the original BMW battery it is now 7 years old and could be intermittently failing internally, low battery power on Mini's causes all sorts of strange electrical glitches and it is not always obvious that the battery is the cause.
Also the various wiring earth points around the car should be checked carefully for good connections to the metal body especially where exposed to outside corrosion in engine compartment, etc, also check for water/moisture ingress into any control modules, connectors.
I assume the main GEM (??) module computer means the BC1 which you say has been replaced, more info on that here:
The BC1 (Body Control Unit) - Forums