Well, I’d say that if the furry elements of your story are no different than putting a hat on somebody, you’re doing it wrong anyway. 
There’s a lot of gray area in how to define any genre or subgenre. And it’s worth remembering that the only real point of genre or subgenre is for things to fit in boxes so readers who like a particular thing can easily find more stuff similar to it in the bookstore or on the website. And if enough people started reading or writing fezpunk, it might well become a thing, a genre of fiction where people’s hats are described in excruciating detail, or whatever. In other words, it’s not so neatly academic and criteria-based, though writers and critics can try to make it that way. It’s more of an organic thing. (Right now, for example, there’s a new category being born called New Adult, aimed smack dab between young adult readers and adults, categorized by having characters who are just out of college, entering the adult world for the first time, dealing with life on their own, etc. Nobody really spearheaded this; it’s just kind of developed from works that got readers’ attention and writers creating more of it in response to an apparent demand.)
My point above, though, was in defense of furry as a subgenre of f/sf, pointing out that to most people, talking bipedal animal characters would fall pretty squarely in “imaginary creature” (and therefore sf/f, which involves imaginary things that aren’t currently possible in the world as we know it) territory outside the fandom (unless you’re getting into political commentary or satire with something like Animal Farm). To some extent, genre is kind of defined by how things look – and last I saw, for example, books with vampire or dinosaur detectives tended to be shelved in f/sf, and not in mystery. The mystery elements might all be there just the same as mysteries starring humans, but the addition of an imaginary creature puts it under the speculative genres anyway.
As another potential facet, I remember hashing this out on the old forums (or in forums somewhere; can’t remember if it was FWG or FA) years ago with ScottyDM, and he proposed the notion that furry is a character-based genre, with the main definition and requirement being an anthro animal protagonist. So there’s that.
Really, though, as much as I have my opinions on this, I have to admit I’ve come to see it as a fairly pointless debate in practical terms. Among the fandom’s readers, I doubt anybody truly cares about the finer points of how to define genre or category and where furry fits into the larger world of fiction. The furry publishers can classify things under their subgenres and sub-subgenres – furry fantasy or furry science fiction or furry steampunk or whatever – and since furry isn’t side-by-side with mainstream fiction in their catalogs, there’s no real reason to worry about where it fits. So inside the fandom, furry gets treated as a category of fiction.
Where it only really matters, IMO, is if you’re trying to sell or explain furry fiction to editors or readers outside the fandom, and people are looking at this thing you’ve handed them (literally or metaphorically) and trying to figure out where to put it. You could probably explain furry as a category and eventually make some headway, but I tend to prefer the path of least resistance, and given that the furry fandom itself grew up out of the science fiction fandom, and that readers who can accept talking dragons and werewolves and the like tend to be more receptive to talking animal characters than readers who prefer things more realistic, that’s still the larger umbrella that makes the most sense to me when I’m talking to someone who doesn’t know what furry fiction is.
