No need to sideload the app, the Shield comes with it preinstalled.

You have to disable location in settings for Amazon Prime video to appear IF you don't stay in the US. That's what I had to do before they rolled out Prime Video worldwide, not sure if it still applies now as the Shield is still mostly US based.

You can install it from the play store.

Do you have the latest shield firmware? It came as a firmware update about a year ago.