I just bought a new bike in Bogotá as a foreigner. It's pretty simple actually--after you pay, they need a few days to prep the bike just like anywhere else.

The dealer prepares the paperwork, which includes plates, title, SOAT (mandatory insurance for personal harm caused while riding the bike) etc. Note that some of the paperwork costs extra--mine was about $300 on top of the cost of the bike, but this might be bike-specific. The dealer gave me the option of filling out the papers on the bike in a different city where they also have a dealership in order to save some money (about $50 if I recall correctly) but you might not have this option depending on where you buy your bike. I chose Bogotá plates because he suggested the bike might face less scrutiny at border crossings.

While they are prepping the bike, go to SIM, which is the Colombian department of motor vehicles, to get your name in the driver registry, called the "RUNT." You just need to bring your passport and tell them "necesito inscribirme en el RUNT." I had to wait for 20 or 30 minutes, provide them with an address, and that was it. But, it could take a few days for them to put you in the system, so if you don't want any delays from the dealership's end you should probably do this before or immediately after you buy the bike.

As for brands, there are a few options in that price range down here but I've heard different things about reliability for long trips. Indian brands Bajaj/Boxer and Pulsar are everywhere, and Colombia imports Chinese bikes under the brand AKT Motos. There are also plenty of cheaper Hondas and Suzukis you can look into, but you might need to look for used bikes to fit your price range.

I'm doing a similar trip to yours--Colombia down to the Argentine Patagonia, back up through Brazil and on the Amazon over to Colombia again. Let me know if you have any other questions about the process!

-Marc