I recommend doing what Shelley suggested.
This wasn't due to AirBnb this was due to someone having your credit card number. If it had been a hotel you would be in the same situation. If it had been $1,000 at Target, same thing.
Report it as fraudulent NOW to your bank/credit card company. They'll get you new card number and reimburse the charge. AirBnb will be contacted to provide any information they have on the person who used it but the bank will not be looking at AirBnb, they did nothing wrong. AirBnb isn't giving YOU this information because its against their privacy policy, it's not their responsibility to go after fraudulent charges, and frankly you're a small fish. It's your banks responsibility and they're big fish.
This article has a ton of examples where fraud (not just identity theft) are examples of exceptions to the Privacy Act: https://www.justice.gov/opcl/overview-privacy-act-1974-2020-edition/disclosures-third-parties#:~:tex....
If you've asked for information about this person, the reason AirBnb isn't providing it is because its against their privacy policy. It is federal law that
Your bank is going to go after that person that stole your identity (yes that is what this is). That person will have to prove that they had consent to use your card, know you, etc. Obviously they're not going to be able to.
If you report it in a timely manner, the bank is very very understanding. If you wait, they're not as understanding.
All AirBnb knows is that the card worked. If someone can provide the necessary information in terms of the name on the card, card number, expiration date, security code, and at least zip code of the billing address (full address vs. zip code can vary by vendor) they're going to accept it.
Take it from someone who has had this happen to them. I also think it's worth considering my comment about Target. Target would tell you the exact same thing.
It's not AirBnb, it's because someone has your CC details, probably your zip code, and at worst your billing address.
Also if you have this card in hand and it's not missing... I would think about how they got it. Where is that card saved? An online clothing store (it me)? Etsy (it me)? Something got hacked for sure if you have the physical card.