CI's official rule is: will not interline if on separate ticket. You will need to pick up luggage at TPE. If you are at an outstation and the agent knows how to interline, they may do it for you if you ask nicely, but technically they won't interline for you.

There is transfer counter at TPE at the sterile area, maybe when you check in at JX they can grab the bag for you (but that's just a maybe).

I believe BR will interline for their partners (and JX is one of them), but not required to. CX interlines at HKG, not TPE.