Chesterton seems to be saying in "vivid representation" what Dulles notes in Newman's Idea of the University and in his other writings:
Rejecting the evangelical notion that faith and reason are mutually opposed, he adopted the view that reason was open to faith and fulfilled in faith. But, conscious of the proneness of human reason to fall captive to pride [Valentin] and self-interest [Flambeau], Newman saw the need for submission to the authority of divine revelation....He recognizes both the value of reason in interpreting the data of experience and the danger of rationalism (p. 144).
Specifically, Valentin has such a high regard for sovereign (not autonomous) reason that he cannot believe that a man who has submitted to the authority of the Church could be superior to him in intellect and in inference. Added to that an [irrational] prejudice about bumbling country parsons, and you have Valentin believing that Flambeau, who has proven himself very wily in his criminal activity, is vastly superior to Fr. Brown. The eyes of the rationalist are simply closed to the deeper truths about things that divine revelation and knowledge of the heart can give to even an ordinary country parson.
Would that it were true nowadays that you could tell whether a man is a priest or not by whether he attacks reason!