For future reference:
The CX check-in desk at TPE was able to check my bags through after a lot of typing.
They suggested that I let them know about different tickets ahead of time in the future. Not sure how that would work...
The two bags made it through TPE-NRT-SYD-AKL, but one bag was delayed at SYD. Apparently, a 90 minute connection gives you hit or miss bag transfer at SYD.