Snape winced and shook his head when he heard what Lupin had said. "He won't have enjoyed hearing that," he muttered. "What Dumbledore does from here depends on exactly how high a priority it is for him to have Lupin in the Order: and that, I don't know.
If Dumbledore went public, it would only harden Lupin's will against him. No, the only way he'd go that route, was if he felt certain he could make it look as though the leak came from a third party, some enemy shared by Lupin and himself: perhaps Pettigrew."