Paid for Amazon Prime on iOS, cancelled auto-pay immediately, and now I have no Prime and no refund.

I paid ₹299 for a month of Prime through the Amazon app on my iPhone. The charge went through Apple. Because I only wanted it for one month, I immediately went and turned off the auto-renewal.

​Well, it backfired. My Amazon account never actually upgraded to Prime, it's still stuck on the free plan.

​I tried using Apple's automated refund portal to just get the ₹299 back, but the site has a problem and isn't working for me right now. Has this happened to anyone else? How did you get it fixed?