It looks like right now ADM1 codes for Taiwan are the FIPS codes, while the ADM2 codes are a partial list of the ISO 3166-2 codes. Would there be any objections using the ISO codes for ADM1 instead?
This seems to be an issue that's only present in Lienchiang (LIE) and Kinmen (KIN), the districts of Fukien (TW-01). If you look at the seat of Fukien (Jincheng, http://www.geonames.org/search.html?q=Jincheng&country=TW), you can see the codes instead of ADM1, 2 and 3. The hierarchy is correct, the codes just don't seem to be assigned to any names