After going through the standard less involved fixes - rebooting both the iPad and Apple TV and resetting network settings on the iPad, and, finally, turning off Blutooth on the iPad (all of which I first went through at Apple’s suggestion), I would consider doing what I did if the problem persists.
First, back up your iPad to iCloud, then connect it to a Mac running the latest version of OSX and iTunes, and then do the factory restore (if you Google “factory reset with iTunes”, you should find instructions.). It sounds harder than it really is - actually it’s pretty straightforward.
Once you’ve done the iTunes reset, set up the iPad as a new device and see if Amazon streaming is now fixed. If not, I suggest making an appointment at the Genius bar at the closest Apple store.
If it does work after you’ve done the iTunes reset and set your iPad up as a new device, erase all contents and settings (under Settings>General>Reset and restore from your iCloud backup. If it works after that, you’re home; if not, time for the Genius bar.
There definitely is something flakey concerning Airplay in iOS 11. It apparently does not affect all iOS devices, so I am guessing it could be a low-level memory leak, which can often be fixed with a fresh install of the OS. That’s why the iTunes reset may fix it, as it did for me. Upgrading to the iOS 11 point releases Apple has done so far — 11.0.3 is the latest as of October 19th — did not do anything for me, but you also could try that before going the iTunes route.
Good luck!