Of course I would have you.
Those words mean something to the Burned Man, something important, no matter how meager the delivery: Gilroy would welcome him. He's not foolish enough to think that the Bishop might be happy to have him, but he could find space in the covenant. With Gilroy's blessing.
Cross considers asking the Bishop how it worked for him, how he was initiated, but decides not to pry. Instead he answers simply: "Branding." The evidence for that form of purification's significance is written, literally, all over his face. "Flames touched me once when I died. They'll touch me again as I'm reborn into death."
It seems the haunt has made his decision. "I leave it to you to decide when it should be done," he says with a deferent tone. He's not sure if there's a specific night on which such things should happen. For a moment, he falls silent, looking reverent and even slightly pleased. He's been thinking about this conversation for a long time and it's finally come to pass. With the desired result. "Thank you for discussing this matter with me, Bishop. I look forward to growing in my faith with you." Whether or not Gilroy believes him, he means it.
But then he snaps back into business mode. "If you have a few more minutes, I have some other matters to speak to you about." More than a few actually.