How can Pope Francis allow the Communist Chinese Government, which promotes atheism, to appoint Catholic bishops?
Your question has an error, as technically Pope Francis appoints the bishops in China, not the Chinese government. Historically, the consultation with the Chinese government in the appointment of bishops is not a novel arrangement; such consultations have been done with numerous governments, including European ones, over the centuries. The Church as a worldwide organization, works with many different forms of governments, dictatorships, democracies, monarchies, etc. Not all of them have a warm, fuzzy relationship with the Catholic Church.
The selection of bishops has typically, in recent years, involved numerous groups of people proposing, screening, and promoting candidates for the episcopacy. In the U.S., lists of potential bishops are refreshed every three years. Historically even the local population had a voice, as in the case of St. Ambrose of Milan. In some areas, the local priests select the bishop, whose name ultimately is sent to the Pope for confirmation. Keep in mind that there are about 5,600 bishops around the world; in the US alone, there are over 300 bishops. It takes an ongoing processing and lots of peoples’ input to be constantly providing names to the Vatican. The Pope, as a rule, in selecting bishops, goes by the recommendations of a myriad of others, who are responsible for vetting the candidates, as he decides whom to appoint. He virtually never knows those proposed to become bishops and never meets them personally.
Is the system perfect, no, as recent scandals can attest. However, remember, 1/12th of the original bishops, picked by Jesus himself, were rotten. Thus, it is more important than ever to pray that God will give us good shepherds, and to pray for the men themselves who are selected to lead the church, regardless of how they were selected.
You may want to check out this blog regarding the selection of bishops.
Countenance of God 