How are you sending the file to your Kindle? AZW3 should work if you're connected to your computer via USB and dragging over manually or sending over via Calibre, but I've seen it not work if you're trying to send it to the email addy associated with the kindle. For that, you are correct, MOBI is usually the preferred option.
It should still work if you transfer it over USB, but if you're trying to do it all wirelessly, simply download the EPUB and convert it to MOBI, and you should be good to go.
