But if they operate in a diocese, they do so under the auspices of the local bishop. If forced to, the local bishop has the authority to shut them down.
Very true.
But I would like you to remember the story about Catholic Healthcare West and Bishop Thomas Olmstead in Phoenix. (He declared that St Joseph Hospital could no longer call itself "Catholic" after performing an abortion...and excommunicated Sr. Margaret McBride, the "ethicist" who approved the decision to do so). To prevent the problem of meddling bishops in the future, CHW changed their name to "Dignity Health" and removed all charter references to them being "Catholic"...rather that they are in the "Catholic Tradition"
See
Dignity Health's statement:
In 1986, two congregations of the Sisters of Mercy joined their ten hospitals together, forming Catholic Healthcare West. In 2012, we changed our name to Dignity Health to better describe what we stand for. Dignity is something everyone is born with. To us, "Dignity" means showing respect for all people by providing excellent care and helping them lead healthy, meaningful lives.
Today, Dignity Health is the fifth largest health system in the nation with 39 hospitals across three states–22 of which are Catholic-based facilities. The mission and values we were founded upon remain the same. Through teamwork and innovation, faith and compassion, advocacy and action, we endeavor every day to keep our patients happy, healthy, and whole. Hello humankindness.
Also think about the Catholic Universities (actually private institutions that were formerly run by religious orders)...they are now almost universally private nonprofit corporations that have lay boards with, perhaps, one or two religious on the boards of trustees. There are any number of examples where universities have defied their local bishops (think about the many controversies with Notre Dame in the past 8-10 years). Consider Marymount...that divorced itself completely from the Church rather than uphold Catholic morality teaching.
Bottom line: if push comes to shove, you will be very, very shocked with the number of these nonprofit organizations that will fold...particularly when faced with loss of federal funds or fines. Not all, certainly, but a whole bunch will go the way of Catholic Healthcare West...