Why doesn't the Betrayer believe in what the Prophet wants?

What the Betrayer sees is the effects of the peace that the Prophet is offering. Because, eventually, the Prophet's message will infect firstly the entire human population of the world, and then spread further to affect every living thing on the planet. And the Betrayer believes that the world cannot exist without balance, that by taking away the balance, the world will eventually be destroyed.

It isn't that the Betrayer doesn't believe in world peace, but that the Betrayer really doesn't want to see the world end because of it. Eidolon City is an attempt to maintain the balance - to take everything which is disappearing from the real world and keep it somewhere, to maintain the balance that way.