Stripe said we lost the dispute because we didn't provide the customer's driver's license. (What?!) We persisted and at this point contacted the customer, who readily admitted that he was still using our software and was happy to pay for it. Unfortunately, even this was not enough for Stripe, who refused to reverse their levy on us. They claimed that someone had stolen the CC from its true owner, signed up for our software, and was emailing us from a fraudulent email address.
At this point, the customer (who was really nice) called their CC company several times to say that it was all a misunderstanding and that the charge was justified. But Stripe refused to accept this as proof. They might not be as bad as PayPal, but they're not saints either.