What you really need to do is root your Fire and install Google Services Framework and the Google Play Store.
Or buy this copy here:
http://www.amazon.com/Moon-Reader-Pro/dp/B004TN7QPG
But you will need another Android device to download it onto, then backup the APK and sideload it onto your Kindle. Since it is an Amazon app, it will use the Amazon verification system which is built into the Kindle Fire (instead of hacking Google Services Framework on to verify Google Play verified apps). Yes, Amazon marks it as incompatible. Amazon lied, and if you sideload it to get around their download ban, it will work just fine.
Moral of the story: Kindle Fire is a lousy tablet for accessing the wealth of Android apps, unless you go to the effort of hacking it. It does make a great way to consume Amazon media, though.