SQ is operating their new 777 on the LAX-NRT route now and I believe it has their enormous flat C seats. Probably the best option for service. Another nice feature of this flight is that it arrives at NRT relatively late (7 PM) which helps with jet lag -- even if you lose sleep on the plane, you can go to your hotel and get to sleep at a locally reasonable hour.

ANA does not have the best seat for sleeping, but they offer one of the only overnight flights to Japan, which leaves LAX at 1:10 AM and arrives at HND at 5 AM one day later... so you can leave Houston after dinner on Monday and be in Tokyo before they open on Wednesday morning, only losing one day. If time is a priority, and if you can sleep on a not-completely-flat seat, that is probably your best option -- any flight to NRT requires you to leave Houston in the morning and arrive in the evening on the next day, so you lose most of two days.

United and Delta have fully flat seats on many flights to Tokyo now, though it's a bit tricky to figure out which planes have them and which don't (Seat Guru helps).