And, to be clear, there are two distinct dogs named Forte. The Forte in Edgar’s present life is simply a stray, not related to the Forte that existed in his father’s day. Edgar names the stray Forte because, though he doubts some elements of Claude’s tale immediately and intuitively, he sees the parallels to the present-day situation as well. It’s also important to keep in mind that in Claude’s version of past events, it is Gar who taught the original Forte to jump into his arms and Gar who shoots him. The photograph that Edgar finds later shows a dog jumping into Claude’s arms—circumstantial evidence that Edgar was right to suspect Claude of shuffling the facts. But why would Claude have lied about original Forte if he knew that Gar was right there and could easily contradict him? Again, the story doesn’t say directly, and it might just be a preemptive denial of a story he’s afraid will come up. But I view Claude as usually working several angles at once, so consider also that one way of gaining someone’s confidence is to tell them a secret, something awful, something that, possibly, the parties involved would not want told. Never mind if it is true or not.
As for why Claude wants to kill, or at least drive away, the present-day Forte, Claude seems to me, as he does to Trudy, a fiercely territorial person—what he has, he wants to keep. By the same token, he wouldn’t welcome competition for something desired but not yet attained, and perhaps at some level Claude views Forte as competition—irrational, of course, but (I hope) plausible. Also, Claude does not particularly like to be reminded of his past. If he sees the stray as such a reminder, he’d have another reason to wish it gone.
To me, what’s most telling about the episode between Claude and Forte is not that he tries to shoot the stray when he gets a chance, but that when he fails, he doesn’t simply lower his gun. He turns instead on something innocent nearby.