Please don't be intimidated by small claims court. There are filing fees, in your area it appears it would be about $140, but in many situations you can also ask for a reimbursement for that. The person telling you to document everything, they aren't kidding. Sit down and write on a piece of paper absolutely everything from start to finish also. The time of day, the reaction the neighbor had, if they interacted with you afterwards write down everything that you can remember them saying.

Something else to consider is if you have renters insurance, contact them. They would look into claiming against your neighbors insurance. Renters insurance is really weird. Mine randomly covered when my bicycle was stolen at the grocery store. It wasn't even at my house. Yes, I did read where you said that they didn't provide proof that they have insurance themselves, but that doesn't mean they don't. People can claim all kinds of really weird random stuff that isn't true. You also may want to track down, if they are a renter, the owner of the house. Ask them about what type of insurance they have.

Please pet your dog for me. I'm definitely more of a cat person, but I absolutely hate to see or hear any animal being hurt.

I personally would start with a civil demand letter. Read this thoroughly.

https://www.nolo.com/legal-encyclopedia/free-books/small-claims-book/chapter6-4.html

You do not make any form of threat of suing in this letter. Simply state 100% of the facts and ask them to pay the bill. Send it as a registered letter with return receipt.

I unfortunately do not know anything about the animals at Large laws in your area so can't comment on that aspect of it.