Sorry to disappoint you but I doubt if it's out of a Cooper 'S'. 22G1128 is the standard 4 synchro remote box housing used on all remote change 4 synchro cars. If it was originally from a Cooper 'S' it would have 9FXEY plus the engine number that the box was originally attached to stamped in place of the M/2833. The only internal difference between an S and non S box is the first motion shaft, the lay shaft, the diff gears and the output shafts, diff side covers and drive flanges. Everything else is the same.
Not sure about what M/2833 means but it probably means its from a Mini K or early remote change Clubman with rubber inner unis.
Hope this helps
RonR