Let's get a few fact clear before deciding what to do:
This tenant is combative.
This tenant has already showed they have no problem lying to you (i.e. they violated a lease which is the same as lying). Therefore, it's likely they will lie to a judge.
You will not be able to replace the carpets until they move out. At that time, if the security deposit is insufficient, you will have to ask them to pay. Probably, they won't. You will then have to sue in court to get damages awarded. Then you get to have fun collecting from a liar who is combative.
Do you have proof that carpet was new? Did tenant sign a move-in inspection stating the carpet was new? If not, what's your plan for proving to a judge it was new carpet that was destroyed? Any receipts from the install showing the date? Also, do you have proof there was a pet? Pictures? Video? Or just a pee stain? Remember, this tenant will lie and say it was 10 year old crummy carpet unless you have solid EVIDENCE--not just a phone conversation--that this carpet was new.
Let's say you go to court, and win a judgment...does this tenant have the $$$ to pay, or will you be trying to squeeze blood out of a turnip? In other words, is it worth your time, money for filing a suit, and blood pressure rising going after a lying, combative knucklehead?
I feel your frustration, but it's worthwhile to lay out the situation before committing your time, effort, and tear to what might be a hopeless cause. My first rental I had brand new carpet in it. First tenants were there only 8 months before I evicted for non-payment. They TRASHED the carpet. Destroyed it! It looked like they brought in wheelbarrow loads of dirt, dumped it, then danced on it. I was able to get a judgment for rent owed, but by the time that happened they were gone and their bank account closed. I never was able to track them down for the carpet damages. The security was barely enough to cover the lost rent.
I learned my lesson: get rid of carpet. Too expensive, too nasty, too easy to be destroyed. Consider cheap laminate, Allure vinyl plank, refinished hardwoods, ceramic tile, or VCT. So strong, very durable, easy to fix one small spot when needed (except laminate).
I learned a second lesson: you must get inside your rentals every 90 days. Find an excuse. Changing furnace filters always works for me, or the "quarterly maintenance inspection" outlined in my lease. You cannot wait until month 11 to see what your tenants are up to, or else you get situations like this one.
Sad to hear about the loss. Best wishes in your attempts to set it right!
--108.250.xxx.xxx