I dont think Taiwan cares. Leave Taiwan with the same passport you entered Taiwan with. Enter HK with whichever passport you want to enter HK with (what ever more convenient for HK entry) and then return to Taiwan using the other passport you want to use. Carry both passports with you so that you can show where you came from if immigration asks. You also have your boarding pass. Tell the immigration officer you want to enter with the country with the passport you want to enter with. Its not a biggie that you have more then one passport.
Just that you must enter and leave Taiwan with the same passport. Also whatever passport you enter with you are subject to the laws pertaining to that passport.
Say you entered with an American passport , you may not be able to work in Taiwan without getting all the necessary paper work being an American citizen. If you then work using your Taiwan ID card (if you have that) then you may have violated the rules regarding work for an American citizen and thus you would get deported. Which only means that then you can return with your Taiwan passport but not with your US passport.
I think thats the only concern. That your activities on the island need to be related to the rights as a citizen of the nation whos passport you used to enter the island with. If you cant work on Taiwan as a citizen of that nations passport, then you cant work even if you are also a Taiwan citizen.
Far as I know.