But a more basic issue in the Peterson unabridged version of the encounter is that he seems uncomfortable. This is in contrast to the YouTube broadcasts of his classroom sessions, his stage lectures, or even his interviews with hostile journalists, where he's fully in charge, and although he often adopts a troubled or working-through-this persona, he's comfortable -- he's ready with answers even to angry politically-correct journalists.
I think the problem is that although Bp Barron is not Thomas Aquinas, nor even Fulton Sheen, he's Peterson's intellectual superior, and Peterson knows it. To a certain extent, Barron is being polite in asking the question in the title here.
I mention the Fringe TV series here fairly often, and I'm convinced it's an important cultural document, however unintentional this may be. In part, it's a sendup of Harvard in the character of Dr Walter Bishop, whose imaginative antecedents clearly include such Harvard luminaries as Ted Kaczynski, Kaczynski's tormentor the psychologist Henry Murray, Timothy Leary, Henry Kissinger, B F Skinner, and even the Harvard dropouts Bill Gates and Mark Zuckerberg.
Jordan Peterson, who taught at Harvard for five years, isn't all that far from this territory. The problem is that Peterson's intellectual foundations are wacky, something the Fringe series's creators seem to have grasped on a deep if unconscious level, but which so far has escaped Peterson himself. He begins with a flawed set of a priori principles that come from 19th and 20th-century psychology, such as Freudian drives and Jungian archetypes, and he tries to apply established principles of reason that date from the Greeks and the Catholic Church, which clearly came to him later in his career.
But having built his career on Freud, Jung, and even Skinner, he doesn't quite know what to do when principles of reason want to supervene. He has to bend himself into pretzels to show how Genesis or Pinocchio somehow illustrate principles that are consistent with his sanitized version of Freud, Jung, and Darwin, when a better approach would be to eliminate the unnecessary detour through modernism. I think Bp Barron, if he applied himself to that task, could dismantle Peterson with little effort, but he's got a different pastoral goal. I think at some level, Peterson understands this.
Bp Barron himself, in his own account of the discussion linked above, finds a "winsome" quality to Peterson, which I think also appears in the character of Dr Walter Bishop. The two aren't all that far apart.
On the other hand, the encounter I'd like to see would be between Peterson and Edward Feser. Bp Barron has reasons not to dismantle or humiliate Peterson; Feser would have no such scruple. But for effective evangelization and catechesis, something like that is going to have to take place.