To check what they have for sale I'd send colvyn y colvyn an email, they sell to foreigners fairly frequently and even buy them back when you're done.

As to crossing Chile to Peru it's generally not the way people go, and I'm not 100% sure if it's possible as a foreigner? There's some info on it if you search the forum. Generally the way to go is through Bolivia and into Peru from there, I recommend the crossing near Lake Titicaca, it's a nice ride there and a much more relaxed, small border crossing.

I've only had 1 problem crossing borders, and it was my fault. My paperwork wasn't in order and the documents I had weren't 100% legit. But, they were very nice about it, they told me how to get it sorted, and even walked me back to the Bolivian border and asked them to let me back in to get it sorted. When I returned with the correct papers all was fine. If you're the owner of the bike and you've got all the paperwork in your name you won't have any issues. My issue was that I bought a bike from a friend in Bolivia, but he hadn't had the correct papers from the original owner back in Chile. Once I'd gotten the correct papers from the original owner (luckily I knew her, sort of) then it was all good. It was more of a case of, I might as well try while I'm so close to the border, and if not I'll get it sorted after.

As to whether there's a "golden plate" to travel with, no not really. If it's Chilean then you need the decleracion jurada, that's about it. People buying Peruvian bikes have had trouble leaving Perú lately but there's a workaround for that aswel.