Another option would be a quick pop to Argentina. Then you´d do a Chile - Argentina-Bolivia-Peru-Chile trip and as far as I know, Peru´s got no problems when you´re entering the country. They just have this law or somthing about bikes with Peruvian plates leaving Peru to Bolivia.
On the other hand, if you´re renting bikes, it should be the rental agency´s responsibility to ensure all your paperwork´s fine. But then again, it´s South America...
P.S. Buying a bike might really be a better and cheaper option. The cheapest one being in Bolivia; another good thing, if you buy a new bike, they sort your license plate and ownership docs the same day. I bought my bike in Nazca, Peru, and it took them 2,5 months to sort out my license plate and papers
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