I think you want to look for award tickets to NRT (Narita) and HND (Haneda) - The two international airports that serve Tokyo.

From LAX, the airlines with non-stop services include Singapore Airlines, United, American, Delta, ANA, and JAL. (If you're unable to find awards on your preferred dates, you could expand this greatly by allowing connecting flights - and expanding your search to include other gateway airports in Japan - KIX (Kansai), NGO (Nagoya), and FUK (Fukuoka) )

How many of you are there? Two? If so, I guess you should have enough Delta + Amex + Chase points for two business class round-trip tickets. My guess is that's the only way for you to get a biz. class / flat bed award. Delta flies LAX-HND. The folks on the Delta forum can certainly give you tips on using the miles in the best way and exchanging Amex points for Delta Skymiles etc.

Another off the wall possibility is to use your Delta miles for a one-way award with Delta. And use AMEX / Chase to get an award with another airline for the return leg. (e.g. JAL from KIX to LAX - open-jaw style approach).

Hope this helps. I recommend you check out the Delta forum.