I'm having the same issues trying to book LAN tickets. Everytime I find flights, I try to make the booking and I get that "they sold out while you were booking" message. Meanwhile, if I go back and search for the flights again, they are still there! When I call BA, they see something completely different! I don't mind working for my award tickets, but it's really hard when you are trying to work with a system that is completely incorrect and unreliable.
I wonder if LAN is the problem? I used Avios to book on Cathay Pacific earlier this year with no trouble.
Has anyone actually booked a LAN ticket on BA.com or have you all aways had to call and have an agent do the booking?