Look for a build code on the radiator shroud.
From mid to late 1963 the 850s body number moved to the radiator shroud with a build code stamped M2/15 in a circle.
Attachment:
850 Body number.jpg
Earlier 850s had their body number hidden under their ID plate. I am not sure of the changeover to the radiator shroud but it was between car 30295 and 40060.
Attachment:
YMA2S1 30295 has body 30357.jpg
Being 5 digits it shouldn't be a Cooper but an extra digit may have been added by a previous owner doing a dodgy. The later Coopers have M2/50 in a circle.
Attachment:
Cooper Body number.jpg