I would hesitate to recommend either a vintage Vespa, or anything with fuel injection. I have an '09 Genuine Stella, and it is a great scooter, very fun to ride, and fairly rugged. It's weak point is the engine. The Stella is a modern made in India version of the P series Vespa. And while they did a great job building the body/frame, the mechanicals are just not that great. I wouldn't even consider taking it on a long trip. Many of the parts for vintage Vespas come from the same company that makes the Stella. In fact, many vintage Vespas have these Indian made engines in them. I was told by a local vintage Vespa shop that it was possible to build a reliable engine for it, but it would not be cheap.
As for fuel injection, I had an '85 Goldwing LTD with fuel injection, and it was a nightmare. It was in good solid shape, but the electronics were constantly failing, and leaving me stranded. I gave it up and got a '95 model with carbs, and have had no problems.
Modern engines with modern carbs are virtually bulletproof. Carbs developed to the point where they were almost completely reliable (unless you left old gas in them for a few months, but then a good cleaning would fix that) when fuel injection came along and brought it's own problems, usually a great deal more expensive than carb problems. Electronic fuel injection is still in it's infancy, and it will likely be a long time before it is as reliable as carbs. A lot of the problems are caused by cheap third world made electronics. If they were properly made, they would probably be reliable, but way to expensive.
So I would recommend a late model scooter with a carb. Vespa would probably be the best brand, it would likely be more common than an Asian scooter. You will not need a lot of speed in South America. But you will need something that will climb mountains, and there is where the smaller CVT scooters come up short. My 150cc Stella with a manual transmission will climb anything, my Zuma 125 and Vino 125 just bog down. Their CVT transmissions are geared way to high for climbing. I suspect your LX150 would do fine on level ground and slight grades, but you will need either more power or lower gearing in the mountains. I wish someone made a modern scooter with a manual transmission, or at least a wider range CVT, but they don't, so I really don't know what to recommend.
2002 Vulcan 750, 2013 Royal Enfield B5
2001 XT225, 2009 Genuine Stella
2012 Zuma 125, 1980 Puch moped