I'm not sure if the pdoc trick with calibre works on the Fire. I never use this method.

However, the Personal Documents cloud syncing method by emailing a mobi to your Kindle so that it can be placed in your Archive and thus treated like a book purchased via Amazon is not yet supported on the Fire. The Fire cannot access archived Personal Documents nor can it sync archived Personal Documents. At present only Kindle E Ink readers can do this. We are still waiting for this feature to be extended to the Fire and the Kindle for Android/iOS apps.