Wirelessly posted (BlackBerry8530/5.0.0.601 Profile/MIDP-2.1 Configuration/CLDC-1.1 VendorID/417)

If you can get yourself to and from LAX on your own, BA will be a good value, as long as you use only CX for the trip. BA has four different award charts, depending on whether you're flying BA only; one partner only; BA plus one partner; or two or more partners. Be sure to check BA's one-partner-only award chart.
If there should be another 40% bonus promotion for MR transfers to BA, so much the better.

There are some fuel charges when redeeming BA miles for travel on CX, but I believe that they're not as high as the surcharges when redeeming on JL.

And since both AA and BA offer one-way awards, you could use some miles from each program to make a roundtrip. Note, however, that although BA's awards will permit an en-route stopover in Asia, AA redemptions only permit an en-route stopover at the North American gateway. So perhaps use BA miles for the trip to BKK, with a free stopover in HKG, and use AA miles for the return, making LAX your "stopover." Then, depending on routing rules, you might be able to continue on -- weeks or months later -- from LAX to Florida, or the Caribbean, or maybe even Hawaii!