Thanks Kovid! I just love the way you support Calibre.
I created a User Defined plug-in according to the instructions and disabled the Kindle Fire plug-in. I specified mobi as the only format because that's all I want to transfer to the device, I de-selected 'Use subdirectories' because I don't think the Kindle Fire can work with them, and set Main memory folder to 'Books' because that's where the device stores them.
After re-booting Calibre the Kindle Fire wasn't auto-detected, but I could connect to it using Connect/share > Connect to folder. The books I downloaded from Amazon are now visible when I select 'Device', and I can save them to disk (in their DRM version) OK.
Unfortunately when I try to transfer a book using 'Send to device' it doesn't appear on the device, either in the Calibre GUI or when looking into the device file structure using gmtp.
Can you help me get further please? I guess I could send books to this device by email or downloading them via Calibre Server (which I have set up - a really neat program!) but I would really like the same ease-of-use as I have with my old Kindles.