At the time, everyone thought that Sirius betrayed James, and murdered Pettigrew and several muggles. Lupin would have believed that he alone knew that Sirius was an animagus.
What reasons might he have had for withholding this information from the Ministry or Dumbledore? Lingering doubts/loyalty? Fear of persecution by association?
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.