How would "The Black Cat" change if the second cat never appeared?

As with a lot of Poe's stories, the narrator in "The Black Cat" is unreliable, so we can't take anything he says at face value. One gets the impression that this deeply disturbed individual would've killed his wife anyway, with or without the presence of the second cat. With someone this psychotically deranged it doesn't take much, if anything, to trigger an outburst of murderous violence.
Nevertheless, the narrator has shown that he's not prepared to accept responsibility for his actions. That being the case, he needs to find a convenient scapegoat—or in this case, scape-cat—to take the blame for his actions. From a legal standpoint, the narrator will almost certainly be absolved of responsibility for his actions when his case comes to court. But if not, he will have no hesitation in attributing blame for the killing of his wife to the unfortunate second cat.

It is the second cat in Poe’s story that the narrator blames for the death of his wife.
Pluto, the first cat, is killed and then the narrator’s house burns down. When he sees a nearly identical cat at the tavern, he is compelled to take it home with him.
Slowly, the narrator begins to despise the second cat just as he grew to hate Pluto. As he and his wife descend the stairs into the cellar of their new, shabbier home, the second cat nearly trips the narrator. In a fit of rage, he swings an axe toward the cat, but his wife intercepts the blow with her arm. Even more enraged, the narrator instantly “buried the axe in her brain, and she fell dead upon the spot without a groan.”
The narrator inadvertently murders his wife, whom he professes love for at the beginning of the story, because of the second cat. One might speculate that without the second cat, the narrator never would have murdered his wife.
On the other hand, he says that the murder and concealment of her body “troubled [him] but little,” meaning he felt little remorse for killing his wife. Therefore, one could say that even without the second cat, the narrator eventually would’ve been driven to kill again.