I think you should contact the people managing the reservation systems and find out how non-ASCII names should be translated to ASCII for their system. You really do not want customers refused boarding a plane because the name on the reservation does not match what's on their passports.

There are transliteration systems to map non-Western names to a Western alphabet, but they typical focus on one language, aren't unique (different newspapers write a Russian name differently), and change over time (China's capital went from Peking to Bejing).

Note that transliteration can have interesting effects. Paris written in Chinese is 巴黎, which transliterated is BaLi. Now you don't want to send your Chinese customers who booked a romantic honeymoon in Paris to Indonesia.