I forgot about this part:

Lupin’s face had hardened, and there was self-disgust in his voice. ‘All this year, I have been battling with myself, wondering whether I should tell Dumbledore that Sirius was an Animagus. But I didn’t do it. Why? Because I was too cowardly. It would have meant admitting that I’d betrayed his trust while I was at school, admitting that I’d led others along with me … and Dumbledore’s trust has meant everything to me. He let me into Hogwarts as a boy, and he gave me a job, when I have been shunned all my adult life, unable to find paid work because of what I am. And so I convinced myself that Sirius was getting into the school using Dark Arts he learnt from Voldemort, that being an Animagus had nothing to do with it … so, in a way, Snape’s been right about me all along.’

He didn’t want to admit that he had betrayed Dumbludore’s trust.