Assuming this is all on CX and that I understand correctly that you are doing HKG-TPE-HKG-JFK...

When you check in in HKG CX will give you all 3 boarding passes (or you can do online check-in and print them), and they can check your bags simply for HKG-JFK. There is no need to send your bags to TPE and back. Then at TPE you simply need to go through transit security at TPE. You really don't need 3 hours at TPE to do this - 90 minutes would be plenty and a same plane turnround is generally no problem.