While I have never had this happen to me, alliance carriers should be able to issue boarding passes when you check in. EVA issues boarding passes for connecting flights but I have yet to get one connecting TO EVA.
If it is one reservation, you can for sure check your bags through to TPE. Show the stubs to EVA at LAX - they record them and try to make sure they are onboard.
They might also have a dedicated line at LAX for people in your situation. It will move much faster than those originating at LAX.