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That's an interesting question, because it shows how much of a complex character Lupin is.
Lupin lost his three only friends in the same period, that must have been traumatic to him. He probably also felt bad having abused Dumbledore's trust during their teenage years.
Lupin is definitely a complex character. I just got to this passage on my current reread:
He thought for a moment of telling Lupin about the dog he’d seen in Magnolia Crescent, but decided not to. He didn’t want Lupin to think he was a coward, especially since Lupin already seemed to think he couldn’t cope with a Boggart.
I wonder if things would have gone any differently if Harry had told Lupin about the dog he'd seen.
Disclosing this would mean disclosing the same about Jamed and Peter, and I imagine he didn't want to do that, so that they could be remembered in the best possible way.
Besides, Dumbledore took him in Hogwarts despite his condition twice - he wouldn't want to tell him his trust was misplaced.
Excellent point about Lupin not wanting to break Dumbledore's trust in him. Not necessarily the morally "right" thing to do, but the reasons are sound. Fortunately, it (mostly) worked out in the end, as Sirius was actually innocent.
It's been too long since I last read it, but I think it felt more like he didn't want to do further harm. Sirius was his friend before and the perceived betrayal did change that but not momentarily, I think.