Normally, a citation is comparable to a traffic ticket. You will need to file your own claim to get compensation for the veterinary bills. You can try sending them a demand letter by certified mail and asking them to turn it over to their homeowner's insurance. If that is unsuccessful, file a claim in small claims court. If they continue to threaten you, you may need to file for an anti-harassment order. You may want to consult a local attorney about this. I suggest that you get a copy of all of the records - animal control report and police report to help you in your own claim. If you wait until after the citation is paid or a finding of guilty in the citation hearing, you will have a stronger argument.
This is not to be construed as legal advice. I do not have an attorney client relationship with you.
I agree with my colleague. Send the neighbor a letter, with delivery confirmation, and ask for payment of the vet bills within X days (include copies of the bills). If they do not pay within that time period then file a claim in small claims court.
As for the threats, speak with the police. You may also want to talk to an attorney regarding your options. Your local and state bar associations can give you a referral.
I hope your dog will be okay.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.