Hope that this info is still valid since it’s been more than one year since I did the whole processes. In short, I bought my first car in Taiwan last year (without having a valid local nor international license). I am not sure if it was legal, but the previous car owner took care of everything. I only needed to provide ARC and a secondary ID card to have the car transferred under my name.
After I got my car, I returned back to my home country Indonesia for vacation while also getting my international driving permit. The process was very simple and easy. Once I got back to Taiwan, I immediately registered my license with the local DMV office. There, they gave me a piece a paper stapled on my IDP (some people refer it as a IDP visa) as a proof that my IDP can be used after 30 days of my stay in Taiwan. Note that if you don’t have your IDP registered, you can only drive with IDP for 30 days upon your arrival in Taiwan.
In order to get the IDP visa, you only need to bring your ARC and/or passport (better to take both of them) and some photographs. Bring it to the counter and they will know what to do. The whole process was very quick, like 10-15 mins include waiting time. I didn’t need to pay anything for that but yesterday I read somewhere that starting from April 2019 the government needs the applicant to pay 150 TWD (correct me if I am wrong) for the application fee.
Using that paper, you will be able to drive until your ARC / Visa date of expiry up to maximum one year. In my case I was only got few months to match my ARC expiration date. However, after I got my new ARC, I can re-apply for IDP registration and got one year extension.
Give it a try and let me know what you think.