Hello everyone,
I'm afraid this is yet another very late reply from a long-time member and first-time poster. If it helps anyone out there, here is an original CMP cap dated 1937. It illustrates many of the features mentioned above, including the red cover, heavily polished chinstrap and a correct badge and buttons.
The last image shows how it might have been worn on duty in Egypt or the Middle East early in WW2. Sorry about the lighting - it was way too sunny but it suited the subject at the time.
Matthew