They will send it to collections and / or a collection lawyer most likely.
I will simply play the role of the devil's advocate here, without really arguing that you did the job and that you were cheated when the buyer questioned the charge. But it is not that the money was virtual, PayPal ended up losing a few thousand dollars in this transaction. They must comply with a series of rules, most of which are established by credit card issuers, on how to handle disputes.
If this buyer was in your area, you can probably take them to the small claims court. If he can show proof of all the work and that he was swindled out of $ 6000, it would not be a hard case to win.