Preparation is the key to a trip like this. If you think a modern Vespa is best try removing the rear wheel on either a GT200 or a GTS... split the exhaust (spare bushing anyone?) and struggle with the monstrous rear bearing that is hard enough to remove in a workshop. Change gear oil? No access without removing the rear wheel! The engine oil change is easy enough but remember to bring spare filters. Spare belts? Know how to change them? The air filter?
On the road the P200 requires no spare parts, even the air filter is a mesh job that needs to be washed from time to time and the filter-free gearbox takes half a pint of common-or-garden 30 motor oil every 4,000 miles. all that needs is a turkey baster and 15 minutes. The wheels are interchangeable with the benefit of carrying a spare. the bike has gears so picking your way on rough roads is easier using the clutch than controlling CVT. Plus they can be bump started in a crisis. Old Vespas are everywhere and they really are simple and well known.
I think its important also to bear in mind qualities that are beneficial in the first world on a time restricted trip may have little bearing on an open ended trip off the beaten motorcycle tracks. To get from jacksonville to San Diego fast a GTS would be comfortable and easy. Go to Scooterwest on arrival and have the rear tire and fluids changed. Ride a thousand miles to Cabo and back and do the same again before riding home to Florida. Done. A journey of 4,000 linear miles to Panama could take 10,000 miles to cover. Better with fewer parts and less need for organized facilities.
HOWEVER an old Vespa will need to be prepared for a journey like this, in most instances. A dual sport would be a good idea but apparently the OP doesnt want to shift a scooter never mind a motorcycle. On the other hand, putting the cart before the horse we know nothing of her experience in Latin America. maybe she has friends in all major cities with modern vespa dealers. There are dealers in many countries, not in every city though, which can be a drawback.
I once had help decoking a P series vespa in a roadside taller
and the owner knew my Vespa had no head gasket, a fact I learned with some surprise 30 years ago in Mexico.