I am really frustrated trying to book JFK-LIM on LAN with AA.

BA and IB show availabilty.
I call AA and the agent doesn't see it.
I go back to BA and it's gone again.
Then I go to IB and there it is, all ready to be booked.

I'd much rather book it for 30K AA than 50K Avios.
But calling AA takes forever, and I don't want to do it every time only to be told there is nothing there.

Help?