I have seen problems with the Fire not updating the LPR setting to Amazon, but they seem to have cleared up since the 6.2.1 update.
Also, you do not have to buy your books from Amazon to have LPR work across devices. As long as you use Calibre to properly set the metadata and do the side loading for each device, it should work. (I've got my Fire and my KK 3g sync'ing my library books that started life as Adobe DRM'd ePubs.)
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