The page below says
Lose your ARC in other country:
- Those who from the countries which are eligible for the visa exemption program, you can apply for ARC after you arrive in Taiwan.
- Those who need visa to enter Taiwan, please apply for resident visa abroad.
http://international.cgu.edu.tw/files/11-1016-2886-1.php?Lang=en
If you lost ARC in Taiwan, you can apply for reissuance. It may be worth to ask TETO or NIA whether you need a resident visa, or visitor visa or entry permit is ok.