After selecting report a problem you then have to select which purchase there was a problem with. There may be an arrow to select the particular purchase then a link on the next page that says "report a problem" next to the purchase. In my situation I did not need to select my purchase with an arrow, as there was a link on the first page that said "report a problem" next to my rental.
I originally purchased my rental on my ipad and intended to stream to apple tv (my apple tv is new and I stupidly just wanted to try out the streaming feature in this situation) When this did not work, I then rerented the same movie on apple tv and reported a problem with my first rental. I just had to give a small explanation for why I requested the refund. Its been less than 24 hours since this situation, and they have already responded to me that I will be refunded the $4.99 for my first rental. Hope this helps.
The exact same error happened to me. I rented "I bought a Zoo" on my iPad (new) for watching on a plane ride from Cincinnati to Houston. The trip wasn't long enough for the movie to finish. After arriving home I tried to AirPlay the rest of the movie on my new apple TV (1080p version), and I get the unauthorized error. I've tried restoring factory settings, re-entering my iTunes account, power cycling, viewing other movie content on the apple tv (movie trailers) and nothing works. I can AirPlay other stuff like utube videos.
I called Apple support and was on the phone (hold mostly) with them for over an hour. They checked my iTunes account, etc., and pronounced that there is nothing wrong. In the end the support person said that there may be a limitation of viewing a rented movie on the iPad with AirPlay. She said that she would continue checking and call me back.
I'm convinced that it's a bug in AirPlay or the new iPad. Very annoying.