I don't think they would even let you check your luggage from LAX to Bahrain. You will probably get it to Taipei, then through to BAH. As far as length, 26 hours is close enough to the distance from London to Sydney. I thought these flights would be hell, but they actually were not as bad as I expected. The only difference is that there was one connection (bkk), and you are looking at 4. Personally, I would just suffer my way through it. If you do decide to stop in HKG, though, check out The Wing. Pretty neat place.
Hyatt Hong Kong. Yikes. I can't even imagine how expensive that would be. Probably more than I make in a week.