Shakespeare was attacked, critically in his lifetime for several of his plays, which gives him considerable cache IMHO.
And, yes, there is very substantial evidence, both from his life and his plays, that he was a Catholic. He seems to have liked living on the edge.
Stephen Greenblatt, Will In The World: How Shakespeare Became Shakespeare.
He sure quotes from the Geneva Bible a lot.
Shakespeare was a genius, and could, therefore, have written all attributed to him with the greatest of ease.