Don't forget that not all potential awards show up online. Some partners, you have to call in to see inventory (or take a guess in advance using the ANA tool, etc...), which anecdotally has increased some availability on certain routes, since they're no longer as easy to see, and cannot be booked online.

I vote for BR given these options. They're a great airline.

BUT! For leisure, my personal preferred routing for that itinerary would actually be BKK-ICN-LAX. You can take OZ or SQ across the pond, with TG or OZ on the intra-Asia leg, the latter of which is even better than BR (IMHO). ICN-LAX on OZ is often an easy award seat to obtain, and that route is on their A380. BKK-ICN is a red-eye though... But my wife loves that route, because you get in at like 6:30am, can take a shower in the OZ lounge (or not - we rarely do, just to maximize time), and catch one of the free Seoul city tours or just take the train into town to explore on your own. Then fly home late-afternoon / early evening, still getting into LAX around lunchtime same-day. You're trading one last night in a hotel for a red-eye in a J seat, but it sort of "buys" you a whole extra day of vacation. (You'd have time for the "quick" 1-hour Yonggungsa tour as well, even if you took the 12pm SQ flight, which is needless to say, awesome!)

Congratulations & enjoy!