You know that you always get the Android install screen after clicking Install from the Appstore itself, this is because Amazon Appstore is considered as a Sideloader of APKs on the android system.
A way to fix this is to move the Amazon Appstore apk in the "/system/app" (or "/system/priv-app" if using kitkat).
In this way Android will considered Amazon Appstore as an authorized system app with install permission, and will allow it to install and update APKs without showing the Android install screen. You can also uncheck the "Unknwon sources" setting to keep your device secure from dangerous apps while still being able to use Amazon Appstore.
I have been using it like this since sometime and it works great.
If you like my tip, leave me a thank you
Enjoy a quicker Amazon Appstore experience...