Yikes! I started on this a few days ago.
There are several themes in the Godfather 3.
Only 1 ties it to the Vatican/Pope.
Here's where I am headed for Godfather 3:
There is a connection between the Vatican and the Rothchilds banking family {owl/y}
There is a known connection with the Vatican and the Germany Army/Hitler/Mussolini, also well documented. {Iron Eagle} perhaps?
Godfather 3 plot line
It is 1979, about 22 years since the events of The Godfather II. For Michael Corleoune, the move to legitimacy is complete: the New York crime business has been handed over to Joey Zasa and all elements of the Corleone business empire are legal, non-criminal enterprises. Michael, approaching 60, is now thinking about his legacy. His charity, run by his daughter Mary, has just handed over $100 million to the Catholic Church. Michael also intends buying a large stake in International Immobiliari, a Vatican-run property company. Things are peaceful and stable but then Vincent Mancini, Sonny Corleone's illegitimate son, starts a feud with Joey Zasa. This has far-reaching, deadly consequences, including for Michael's deal with the Vatican.
So this part of the storyline plot is what has to be the dig
"His charity, run by his daughter Mary, has just handed over $100 million to the Catholic Church. Michael also intends buying a large stake in International Immobiliari, a Vatican-run property company."
https://www.theguardian.com/world/2013/jan/21/vatican-secret-property-empire-mussolini
British wartime records from the National Archives in Kew complete the picture. They confirm Profima SA as the Vatican's own holding company, accused at the time of "engaging in activities contrary to Allied interests". Files from officials at Britain's Ministry of Economic Warfare at the end of the war criticised the pope's financier, Bernardino Nogara, who controlled the investment of more than ยฃ50m cash from the Mussolini windfall.