I started over so I could show step by step what I did.
- Uninstall CC
- Erase the SD card
- With ES File explorer, look at the SD card, and in particular Android/data. CC isn't there.

- Install CC. Change no settings.
- Again with ES File explorer, look at Android/data. This time we see CC's private folder.

- In CC, go to Settings / Formats and Folders. Tap the "Default" line at the top. Pick Choose Folder. I see the following, and in particular a Use SD card button.

- I tap the Use SD Card button and see a warning. I tap OK and I see:

- I tap Use this folder. I see an "Are you sure"?. I tap OK. I now see the Formats and Folders screen showing the path to the SD card.

- I press OK to close the Formats and Folders dialog then use the Back button to get to the main CC screen. I check the path Android/data/com.multipie.calibreandroid/files with ES file Explorer. The folder is empty, as it should be.
- I connect as a wireless device and send 2 books to CC, and epub and a mobi. Checking with ES File Explorer I see those books on the SD card.

- Go to book details for the epub and choose "Open book with ...". CC offers me FBReader. Choosing that opens the book.
- Go to book details for the mobi and choose "Open book with ...". CC offers both the built-in reader and FBReader. Choosing the built-in reader opens the book, resetting the reading position to the beginning. As expected, the book is not shown in the Fire's "Books" screen.
As far as I can tell the Fire did exactly what I expected it to do. It offered CC's private folder on the SD card and stored books there without a problem. The only "problem" is that the built-in reader doesn't remember the book, but there isn't anything CC can do about this.