Hey! I had the same plan, and can maybe save you a few headaches!
I arrived in Santiago on Jan 21, hoping to buy a used or new bike. I stayed at Casa Matte(great spot! I learned a ton about bike maintenance just watching people work, and Kristian is the nicest human on this planet

, where I learned of a few issues with Chilean plated bikes from other travelers. therefore i changed my plan. (Which is actually turning out for the best as I don’t really want to be in Patagonia at the height of the season anyway)
Met an American, chilean plates, ownership paperwork- has been denied at 7 border crossings
Met a Belgian- bought a new bike, submitted paperwork to have ownership papers (which can take 3+ weeks to get)...the agency lost his paperwork. He submitted again, and paid-agaib, and they lost the paperwork, again. He is now stuck in chile with a bike he cannot legally sell.
I read that Peru was a good country to have a plate from. I bought a bike and will pick it up in a few weeks, then was contacted by a few people who heaes my plan through the grapevine with their problems at border crossings and a 15 step guide on how to get out. Yikes, this next month should be interesting
Met 2 Americans that purchased brand new bikes in Colombia, had ownership papers in their hands within the day, havent had a single issue crossing any border.
Colombia seems the way to go! I will let you know how leaving Peru goes for me
Im splitting my travels into several chunks after hearing it is possible to store the bike for 6 months in Colombia or 1 yesr in Uruguay, so will come back in Nov for Patagonia.
Hope you get on the road soon’