Just an FYI on the BA.com recommendation.

The engine tends to give phantom space. When Lan First Class is shown as available, this is your indication of phantom space, as LA does not have First, only Business. On very rare occasions, the search will return 7 Biz and 0 First, and all 7 tend to be phantom. I myself didn't believe this one, but 3 different ExPlat agents did not see availability. Overall, space is easy to find come 1 month prior to departure. Enjoy the 787!