This is bizarre. I had ZERO problems booking any of my 20 segments on my RTW, on QF, CX, BA and, yes, AA, including LHR-LAX only about a week before you, as well as SFO-JFK on the AFS. But it does look like 3/31 and 4/1 have no A availability, but there is in the days before and after. So if you have no flexibility to travel, say, a day later, then I suggest you take the BA flight. It's a bummer to lose the miles, but I guess you'll have to prioritize.

In general, I think you may be over-reacting a little bit. Yes, sometimes there will be availability difficulties. But more often than not, those seats will open up eventually. You are on a discounted ticket, and AA reserves the right to limit the number of those seats sold on a given flight. I strongly doubt that they are limiting paid A inventory nearly to the extent as reward inventory.