Guy Betsy, I had intended to ask only for general advice (give up versus call AA RTW versus call LA reservations; confirm in D and waitlist for airport standby, versus extra effort to confirm A in advance) rather than concrete suggestions. Didn't want to burden folks here with my problem specifically so much as inquire about the LA situation generally, probably more useful to others in the future.
But since you asked: I could fly LIM-LAX on Jan 29, 30, 31, Feb 1, 2, or even 3.