Recent behavior that DSP is doing gives me the idea that indeed it's not his cat, and he's very terrible at knowing how to treat a cat, even after a year as he claims. Might be extra careful with it because he's being broadcasted, and it's actually Kats "baby" and doesn't want to inconvenience in even the slightest way(picking it up as an example).
That's most likely why he's so afraid of it. He's hardly around it, it's not his, Kat's protective of it, doesn't really give a shit about it himself, and I'm still thinking it's not been in their possession for as long as he claims. Which is why it only comes around when Momma Kat isn't.