A general rule of thumb is to have as few middlemen as possible. Thus not starting with a travel site like cheap tickets. If you encounter problems, you don't want to make it easy for the different parties to point at the others and blame them.

Another is to make the deal as simple as possible. Just getting the business class ticket in one shot, rather than a two-step (or more).

And both of the above particularly apply when you are a newbie to a type of transaction.

I hope you have flexibility as to dates. There can be many miles difference between one ticket date and another. In both directions. Sometimes you will learn something by searching separately for the trip there and the trip home. Also, do look at any of the LA area airports that would work for you, plus both Tokyo airports.

I only looked at a few dates in June, but I can see Business Class award tickets for 240,000 miles. Just to get you started. Now jump in. If you find a better deal, come back and brag a bit.

I didn't look at the added $$. These can vary quite a bit. Check before you finalize your deal.

Last factoid - if you actually hit the button to accept the award ticket, you do have 24 hours to cancel. I'm only sure of Delta's rules, but with them it means 24 hours and then on to midnight. I'd guess AA is the same, but you'd have to check to be sure.