Okay - first - welcome to hosting. Many of us have had bad experiences while we were learning the platform.
1. Don't care what anyone says - no pets. No Emotional Support Pets. Airbnb's policies are out of whack with federal guidelines that say owner occupied homes and duplexes are not subject to laws on ESA. Plus - you have a dog already. So mark your home as unsuitable for animals due to the presence of a dog on the premises.
2. Here's how this should have worked. YOU DO NOT NEGOTIATE OR REFUND for guest damages. You inbox them in the message center to explain your distress. Then you bill them in the resolution center before the next guest checks in. Upload any estimates on replacement costs and photos. If the guests won't pay, Airbnb will start their investigation and reimbursement process within three business days. The more you can show them, the better.
3. Don't mention pets in your complaint. Airbnb doesn't pay for animal damage. This is guest damage and it doesn't matter how it happened, the guest screwed up.
Vett your guests better and make it clear in your listing that damage is billed at "replacement cost".
However - and this is a big one - it could have been your dog. Dogs are territorial. If the guest dog smelled your dog, he may have marked his territory. But vice versa, it could have been your dog marking his territory. So just say no to animals and ESA citing safety issues.
Sorry for your loss. The guests were required to disclose a dog that wasn't a service dog. You could have rejected them when they showed up with one. Proof that it was not a service or ESA pet is that they left it alone in your house while they worked. Some people are inconsiderate. You'll learn, over time, how to screen those people out by asking lots of questions and creating rules they are required to read before booking..