If they wish to remain independant without a whole lot of pain then any name that uses or implies Taiwan is out of the question.
Due to bribing bullying, greed and coercion there are less than a handful of countries that officially recognise Taiwan as a separate entity. Not many will come to their rescue if they antagonise China enough to be invaded other than pointless pontificating in the UN where China holds a veto.
China in the name pleases China but doesn't suit citizens who support independence.
The chosen name will by its very nature will be political (if it happens).
To please China and traditionalists it will have to have a Chinese flavour without a hint of independance but the other side will not tolerate the new name containing "China" or implying unity with the mainland.
Some reasonable suggestions here of Dynasty and Mandarin.
Dragon Airlines is a clever recognisable chinese flavoured name but does not suggest it is Chinese owned but does not offend the CCP. Unluckily for Taiwan it is already taken.
Some obscure name that doesn't associate with Asia could be detrimental to their visibility and recognition.
It's an airline so needs to engender a feeling of safety reliability and comfort so it doesn't want a name to sound like a Laminex tabled restaurant. A word like Panda is too cutesy/non business like and would probably not please CCP either.
It seems to me words associated luck, longevity and wealth are most warmly received in Asian venues. Golden comes up regularly so does Eternity, Lucky, Oriental and Jade.
They will need to verify the new name does not offend passengers or countries on translation.
Existing trademarks may also provide difficulties.
Will be interesting to see what they come up with.