I had some trouble last week sideloading a native amazon app to our offices Fire Phone so I wanted to share the easiest solution for this installation. Normally with other android devices I add the apk to the root and simply navigate to the file on the phone for installation but it seems the device does not allow for root access.
1.) Connect the device to your local machine which contains the Amazon apk file
2.) Transfer the file to whichever directory
3.) On the Fire phone, install the app APK Installer and simply navigate to the native app and install
Hope this helps,