Sirius attempts to kill a rat in a dark room with five sleeping children. Rats are small and quick, and knives can get snagged in objects such as bed curtains. Overall it seems a terrible idea to bring a knife to a rat fight. Since he already had his wand back at this point, why didn’t Sirius try to stealthily capture Scabbers, rather than flailing a blade around? Was it purely for dramatic effect?
Since he already had his wand back at this point,
I don't think there is evidence supporting this premise. On the contrary, when Harry and friends meet Sirius in the Shrieking Shack, Sirius uses first Ron's wand and then Snape's wand – presumably because he did not have a wand of his own:
"Expelliarmus!" he croaked, pointing Ron's wand at them.
Black had already retrieved Snape's wand from the bed.
If Sirius did not have a wand then the reason he used a knife was simply because that was the weapon available to him.
Some of the arguments:
Sirius escaped from prison without his wand by means of Transfiguration. He was an unregistered Animagus and he escaped through the bars when the dementors brought him food. Therefore, Sirius is wand-less and couldn't cast spells against Pettigrew.
He escaped as he saw Scabbers aka Pettigrew in the newspaper. He knew that Pettigrew was with Ron and rushed off to their dorms. Imagine Sirius seeing the person he believed in his past, not able to say a word after he knew about the betrayal. He was so engrossed with rage that he acted out of impulse as soon as he saw the traitor. He won't be able to capture him and have a talk and kill him. He was extremely angry at Pettigrew that he wanted him dead at any cost.
These may have been the reasons for Sirius' actions.