Well, you could book one of the options you see available now, and then cancel (quite possibly at some $$ cost for redepositing miles) down the line if CX or some other better option opens up. As others have noted, CX does sometimes open award bookings on flights a week or so in advance.

One other thing you could do now is join awardnexus.com, which is free for some initial use by FT members. It allows you to search across several different airline programs to see if seats have opened up. For example, I've occasionally seen CX award seats available by checking the JL(CX) option on that site when BA shows nothing for CX, and vice versa. If you want to take this a step further, you could then pay some $$ for a higher level membership that sends flight alerts if seats open up.

Not sure you've done this yet, but you could also break up your searches on airlines that have F so that, for instance, you're checking for two F and two J, rather than one or the other.

Finally, have you checked award availability flying to SFO instead of LAX and then connecting from there to LAX as part of the award? (You could also do that for other cities that a given Asian airline might fly to, such as DFW, ORD or YVR, though this obviously makes for longer flight times.)

Hope it all works out.