So I ended up booking with China Airlines. They are flying one of their new planes which has three classes of service. First/Business, Premium Economy, and Economy. The Economy fare was $943 r/t and the Premium Economy was $1443 r/t. I opted for the PE for the following reasons:
1. The seats are two inches wider than economy and you get a few extra perks like early boarding, better food selection, and a couple of other minor things like a big/wide armrest between seats so you don't have to fight over who gets the armrest.
2. If you book PE and the second leg of your journey doesn't have that class of seating, they bump you to first class as long as there are seats left in first. I booked early enough to get first on the second flight.
3. And because I will have a first class ticket from Taipei to Palau, I will have access to China Airlines Premium Lounge during the 7 hour layover in Taipei.
Overall, looking at schedules and prices of the various airlines, China Airlines offered the best deal by far. I will plan to report back after my trip to let everyone know how the flight experience went.