Plenty of foreigners have purchased and registered bikes in the US. If you use google to search this site you will see a few discussions of which states this is easy to do it in. In summary, you will have a bike titled and registered in your name so no issues in crossing borders.
The reasons that many choose this route is that the US is an inexpensive place to buy and outfit bikes and it will save you the costs of shipping a bike across, which depending on whether you ship sea or air can range anywhere between ~$1,250 and ~$2,000 USD. That will buy you a lot of in South America!
Once you get closer to picking dates, let me know and I'd be happy to point out any outiftted bikes that are fairly regularly posted on the usual bike selling sites. If possible, it is nice to sort out a bike ahead of time or at least narrow it down to a couple as hanging out in a city and travelling around to look at bikes can significantly eat into the cost savings from not shipping.
Happy planning.
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