Problem is, you can't convince him of this, since he argues by contradiction...So his uniqueness argument is not merely unproven, but unprovable and unnecessary. ROFL
You can come up with a different kind of algebraic structure where there are provably finitely many twin primes, and show his argument fails there. Or you can do like collag3n above and consider teta-primes to show his argument fails. But he doesn't understand those, so he will dismiss it.
By the way, you might be interested in a simple proof of Bael's conjecture, also very hilarious: The Simple Proof of Beal's Conjecture - xkcd