I would not get the ID card until I was sure I am no longer eligible to be drafted. I also have US and TW passports with no ID # on the latter one... About two years ago, I asked the immigration office about the requirements of getting an ID. Basically, just enter with your TW passport, fill out some paperwork and present several pieces of documents AND stay in TW for 366 days consecutively. The problem now however, is that if you stay past 365 days with your TW passport then you have to serve in the military!!! Don't be fooled! Just get an ARC with your non-TW passport and work until you are past 36/37, and then apply for the ID card.
If you are someone who left TW when young and obtained a passport from another country later on and decide to pop into TW for a visit and work, the authorities here may be able to match the name and picture in your new passport with their records, BUT they cannot force a foreign national to serve in their military. However, in their eyes, it is your duty to serve your country, why should you be able to live overseas for so long only to come back and cut in front of everybody? This is why they give you trouble on your ARC, to send you a message of, either serve your duty first or leave, because the current employment situation in TW is not good. I don't think you can circumvent this by legally changing your name because they are now taking fingerprints.