Hi there,
I did a similar trip to you last year however I went straight from LIM to PEM (Puerta Maldonado to see the amazon). Then PEM to CUZ. The LIM to PEM flight goes via CUZ but is the same price as LIM to CUZ (i.e. 4500 avios). So thought it was more worthwhile to goto PEM first then CUZ if that makes sense.
I just did a dummy search on BA. I managed to get flights. Just type in Cuz (it wont recognise it initially). But on the next page select include partners and it gives loads of availability.
Got about 4 reward bookings when I went around south america - all 4500 avios and £5 or so tax!
Heres a screenshot