Very helpful suggestions from everyone as usual! However, I still seem screwed by availability ... they don't even have any L inventory on this route. Aargh. It does seem very strange that I can't pay them for one sector where there is no availability...

Meanwhile the ExpertFlyer help desk has shown me that if I search on LA specifically, rather than OneWorld, I get accurate results. This is a tip for others too (although it looks like a bug to me). I tried again now, and on one date the availability is now consistent but on another, search on oneWorld vs LA does generate different A class results.