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.
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.