Follow-up: I contacted customer service, who sympathized and took down my details, then warm transferred me to the "Recovery Department" where they're responsible for trying to collect for these types of things.

The woman was quite friendly and asked me one question: Did I hit anything with the car. When I told her no, the tire just blew out on the freeway, she asked me to send her an email to that effect, and she'd try to close the case. I sent the email, and put in the details about being in the #1 Club and how I was surprised to get a bill for what I considered roadside assistance.

Within an hour, she responded that the case had been closed.