Leave the Chase and AMEX points in their respective accounts until your travel plans are almost firmed.

Use BA miles to South America- but in some situations AA miles may make sense- for example Peru is counted as Carribean/Central America and is available for 17.5K IIRC. Use the tag-along trick to "add" a segment to Hawaii for free, and then you kill two birds with one stone as mentioned above.

You can also use this trick to Europe (tagging along a segment). If you don't want to visit Hawaii twice you can tag along the Hawaii segment on the South American trip, and on the European trip, tag along another segment to Alaska or somewhere. This works too.

South East Asia is best reached using Star Alliance miles, or AA miles on Cathay metal. Star Alliance offer more options but Cathay is decent if you can find availability. Asiana's always a decent product, as is ANA, and they both are (relatively) generous with their award space.

Good luck!