We tried to debug my issue with Clint and Sighery, it's a problem with the authentication using the pairing process. The keyboard asks for a pairing confirmation that doesn't show up in the ace_bt_cli interface.
I tried to pair another BT keyboard, and it works! 
Now I can play with Textadept on my Kindle!
Another question: how can I change the disposition of the keyboard? I need to set it to French (azerty).
Update: Solved. See below 