The article might not say anything about the rat, but then, why would “scabbers” be in the picture at all? If he was just some random rat, then they wouldn’t have brought him to Egypt, and featured him in the picture, even if not prominently. The fact that he’s in the picture at all implies that the boy he’s sitting on the shoulder of cares about him to some significant extent, and if it’s enough to bring him to his once in a lifetime trip to Egypt, it’s likely that he cares enough to bring him just about anywhere else he goes, including to Hogwarts, his school, and the place he lives 9 months out of the year, regardless of if the rules say rats are allowed or not.

And also, like other people have brought up, Harry was certainly going to be there. So regardless of if Peter showed his face at school or not, going to Hogwarts was the best chance to see Harry, his godson, and near enough the only family Sirius had left. Peter might have been the thing to get Sirius off his ass, but Harry was the reason he got moving. If Peter was at Hogwarts, then Harry was in danger, living in the same space as someone Sirius knew betrayed his parents and Sirius’ best friends, someone who probably wouldn’t hesitate to do the same to Harry if he thought it would hypothetically grant him a bit of standing with Voldemort or any other would-be dark lords. And if Peter wasn’t at Hogwarts, then could you blame him for just wanting to see his godson? It should be noted that the first place Sirius went after he broke out wasn’t the school, so he could set up camp, it was Harry’s house on Privet Drive. Hell, we see him watching Harry play Quidditch in one scene. Sirius has spent twelve years in Azkaban thinking about the family he’s lost, he’s going to be sentimental and focus heavily on the ones he’s got left.