This may interest you.
Out of the curiosity your book inspired I bought the attached item because it's so like Johnson's bracelet. It has faint traces of soldering on the edge so it was probably once an identity bracelet too. I also purchased an example of the host coin which tells me that half the thickness was removed to make the plain face for engraving.
What's instantly striking is that the depiction of the mosque is so similar yet the lettering indicates a different hand involved. The hatching also goes in a different direction. I wonder if these two pieces are evidence for a field workshop in the Baghdad area making these to individual commission? In my one's case the host coin is an Iranian 2000 dinar piece of 1911 but it's the same diameter (28mm) as yours so obviously had been chosen for the same purpose.
That the images are so similar does suggest craftsmen working together and copying a template design. I might have found the building it's based on. The picture attached shows the Sheikh Omar mosque in Baghdad. It's obviously not identical in format but the mosque has the distinctive spiralled minaret and curved palm tree found on the coins. Impossible to prove, but a possibility.
Sadly, Pte F Bates 66396 of the Machine Gun Corps seems to be a soldier with no surviving records so his story for the moment stops there, unlike all the fascinating biographies you uncovered. But you may come across other items made by whoever worked in this little workshop which may eventually tell us more about the when and where and how. These two pieces are too similar not to have been made around the same time in the same place.