As a postscript to this one, I did get through the Bolivian border. It did seem a little touch and go when they discovered that I wasn't resident in Chile, but they seemed to more or less shrug it off and let it go.
The next day I crossed into Peru from Bolivia without incident, though did have to wait 3 hrs for the Bolivian customs to return from lunch (note - don't try and cross the border on a Sunday lunchtime!).
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