Apparently Battle.net's mobile authenticator can be downloaded to the Kindle File. The trick is that it is available via Google's Play site, not via Amazon's app store. I have no idea if this works or not.
http://us.battle.net/wow/en/forum/topic/3595675350?page=3
Here's another convoluted method which only works if you have an Android phone to download to.
http://www.tested.com/news/feature/3179-how-to-sideload-android-apps-on-the-amazon-kindle-fire-in-3-steps/
The easiest method is to use Getjar.com to put apps on your Kindle Fire but Battle.net mobile authenticator doesn't seem to be one of them.
http://www.businessinsider.com/android-kindle-fire-apps-2011-11?op=1
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