Travel with Air China. Benefits to have 72 hours visa-free in Beijing.

Copenhagen - Beijing (direct flight by Air China)
Spend 72 hours in Beijing
Beijing - Taipei (direct flight by Air China)

Taipei - Beijing (direct flight by Air China)
Spend 72 hours in Beijing
Beijing - Copenhagen (direct flight by Air China)

In this case, you can have 144 hours in total in Beijing. You could have a extra bonus travel experiences not only to Taiwan, but also to Beijing and its serrounded cities and areas such as Tianjin, Xi'an, ect.