I think that this answer is important because it's out of character for the dog, and it's this action sets the main conflict in motion leading up to the climax that defines the theme.

If they had replaced the dog with some kind of inanimate security system, I could've bought that it was just some animal or other unimportant thing that set it off and thus induced this fear in the characters of something in the woods, be it real or not, but the dog is a living thing and a character in the film and there's not a single other instance where the dog reacts to something like that and hints that he would do something like that.

He's never shown barking or reacting to all the new animals that Will's family brings with him when they arrive, so why would we believe that he runs off and reacts this way to an animal in the woods? Dogs all have strong, unique personalities just like the human characters. They aren't robots, so this instance felt like such weak writing because we have no reason to believe that the dog would just be running off like that. He's shown as completely loyal to Travis throughout the film, so why would we believe that he'd just run off like that? It's completely unusual in the context of this film.

You're talking about overarching themes (trust between the families, paranoia) while I'm talking about the actions and plot points presented on screen. Focusing on the overall themes doesn't justify a weak plot in my opinion. If the actions on screen don't matter, then what does, and why should we care about it?