But is that likely to be the problem here? After all, you say they DO see availability for the day you want, albeit for one flight only.

I would pick a few days in Apr/May (as a test only) and look for 2 or more award seats on lan.com, and with that information call the EXP line. If they again see only the last flight of the day as available I would know the problem lies elsewhere, and would try to persuade the EXP desk into looking further as to WHY only that flight shows as available. AA cannot contact LP's RM but they can contact (by phone, if you push them) a OW liason person who can in turn provide, hopefully, some answers.