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AA's main partner for South America is LATAM, and you can't look up LATAM award flights on aa.com.

I suggest you go to www.awardnexus.com, click on "more signup options" under the green Purchase button, and sign up for free as a FlyerTalk member (look for "I am an active member of flyertalk.com" on that second page). Award Nexus lets you search using either BA and/or QF (Qantas) sites's for oneworld alliance award inventory (which is bookable on AA only over the phone in LATAM's case). Sometimes QF gives better LATAM results for me than BA does. With Award Nexus, you use the same user interface whether searching via QF or via BA. You could alternately sign up for BA and QF directly, but then you 'd have a totally different user interface at each.

If that fails, the other thing you may want to look at is looking up business class flights for your trip on Kayak. You might be surprised at how inexpensive some business class flights to/from South America (with changes of planes in Central America or northern South America) can be. Those business class seats may not be sleeper seats, but you can get daytime flights often. Though just as in the case of award flights, you might want check other USA airports (like MCO and MIA) in case those lower business class fares are only to/from those cities and not TPA.