ROME—Pope Francis’ recent decision to replace two Chinese bishops loyal to Rome with selectees of the country’s Communist government, heralding his broader moves to reset the Vatican’s ties with Beijing, has drawn cries of betrayal from advocates of the country’s long-persecuted “underground” Catholic Church.

The pope’s actions in China are characteristic of a leader who has repeatedly practiced realpolitik to achieve important goals. But they clash with Pope Francis’ image among many Catholics and others as a defender of...