Lots of options!

Least expensive way to ship would be shared container out of L.A. Find people shipping car(s) where there is extra room for a bike or two in their container. If they are honest, then should be reasonable. (I've seen offers of about $500) More bikes means cheaper for everyone.

You could also ship air or sea from either Galveston, TX or Miami, FLA to Colombia. It's a short hop. Cheaper than going Chile. Break down your bike, crate it up SMALL to save money on Air shipping.

From Colombia you could ride South, see countries you want, then leave bike there, fly home, return in a year or whatever, continue travel. ?
Just a thought.

You could buy a bike locally in Chile. Best buy travelers bike, if it's a California bike, you transfer ownership to your name so you can traverse borders. There are threads on here and riders who've done this RECENTLY.
IT CAN BE DONE but not always strictly legal.

Lots and lots of KLR's posted here for sale in S. America, mostly US registered bikes ridden down by travelers. Some good deals for set up bikes. This could be your best bet.

You could also go through the paperwork and buy a local bike in Chile. This is a bit complex and takes TIME. If you have friends who can work ahead of your arrival, might be OK.

In Chile, anything Japanese or EURO is overpriced and often BEAT. Chinese 250's are affordable but not very good travel bikes IMO.

I'm not a fan of the old F650 BMW's. Lots of issues that can easily RUIN your trip. Research.
Do consider the Suzuki DR650. Simpler, easier to work on, stronger and lighter. If buying a used on is USA, very good deals to be had on new-ish, low miles bikes. Craig's list.

At present I'm waiting to hear from a guy who is trying to fly he and his buddies bikes out of Galveston to Colombia. Will let you know the rate when I hear, as I'm thinking of doing this exact thing. (Galveston or Miami)

Good planning!