I think I'm out of luck as none of the partners shows availability between those airports. Thanks for everyone's suggestions!
A few other ideas:
1. Not every frequent-flyer program website can display award-seat availability on every partner. For example, aa.com cannot display award-seat availability on LATAM, but ba.com can --- if you have a BAEC account.
2. I don't know Brazil. Are there any airports close to RBR with nonstops to RIO?
3. It appears that RBR-RIO cash tickets are available for less than $300 -- at least on some days.
Good luck!