Install the Amazon app, but not the Amazon app from Google Play, the sideloaded one directly from their site, and then do a bunch of convoluted stuff to install the Prime Video player within that app.
It's been a couple of weeks since I did it, but I got it working on my nVidia Shield TV, and I remember the process being something like:
1) Sideload Amazon app APK
2) Sideload Amazon Video app APK
3) Open Amazon app and log in
4) Open Amazon Video app
However, it wasn't a pleasant experience. The version of the Amazon Video app I was able to get working didn't scale well to large screens, didn't work well with remote-based navigation, and didn't seem to do HD. I watched one episode of something, and haven't used it again since.
