KLM is highly regarded in the pet travel industry, and their in-cabin allowance for pets is generous:https://www.klm.com/travel/et_en/prepar ... /index.htm
Please, do not send your cat as cargo or baggage unless you're using a pet travel agent. They are experts who know their way around the system. It takes only a few minutes for the temperature in your cat's cage to reach an unacceptable level during transport.
KLM offers flights from Taipei to many European cities via Amsterdam, KLM-operated. There's a world-class pet hotel at Amsterdam Schiphol Airport where you can take your pet for care and rest during the stopover. You would need reservations and you would also pay extra for it. The benefit is that your cat will get a chance to de-stress, drink water, and use the litter box. They have veterinarians on call if your cat needs help, and you don't need to worry about importing your pet in order to get medical attention.
I don't have experience with agents (I always keep my cat with me in the cabin), but I do know some airlines will not even allow you to transport a pet by cargo EXCEPT if you have hired an agent, and so offer approved 3rd-party agents for the job. This is a good sign, IMO. If you book with them, they should have an agent available for their routes.
Some airlines (United/Delta) offer a premium-type service for pet transport that guarantees of the level of care that will be given to your pet. This is not good enough, IMO, simply because it is a within-corporation service. It would be carried out by baggage employees who are following instructions about how to care for your pet, and though I expect most are well-intentioned and hard-working, the motivational scheme is not all comparable to the first scenario.