You can only enter Taiwan on one passport.
If you enter with your US passport, you will need a visa to stay beyond the visa free entry period for US citizens.
Leave the US with your US passport. Enter Taiwan with your Taiwan passport. Leave Taiwan with your Taiwan passport and enter the US with your US passport.
The tricky part is ticketing. Your name on the ticket must match the name on the passport being used. Sounds simple, but its not when you have two names, one Chinese and one English. Be aware of that when you book tickets.