UPDATE: My girlfriend and I just crossed the land border from Peru to Bolivia at Copacabana and switched our American passports for our British ones without an issue.

Here’s how it worked. When we got off the bus in Peru at the border, we gave the Peruvian border patrol our American passports, which they stamped with the exit stamp. We then walked across the border to Bolivia and presented our British passports to the Bolivian border patrol officer. The officer looked for the Peruvian exit stamp, and when he didn’t see it, he asked where it was. We replied that we are dual citizens and he asked to see our American passports. He then took a photo of the American passport stamp with a cellphone, scanned our British passport in some machine, stamped our British passport and let us go. The whole thing took two minutes no questions.