KVS shows award availability for flight 531 from JFK to LIM on August 24th. It also shows availability for flight 530 from LIM to JFK on September 1st.

The BA website tool is really... patchy. I've found one (and only one) LIM -> JFK flight available on November 3rd, but if I enter that date in the original date select request, it says no flights available. It only comes up if I search seven days back and then seven days forward.