Just had a look at your listing. You clearly state that you have cats. Your guest knew your cancellation policy when he booked, plus you have correctly labelled the lisitng as a private room. I would say that that is why you have a moderate policy, and it is on the guest to read the listing properly.
I would recommend you flag the messages, and contact Airbnb to manage this guest. You are entirely within your rights to not refund and you shouldn't be hounded for it. Your listing is very clear.