Resurrecting this thread to share my solution:
The Fire TV Stick Remote enters some kind of standby mode, which disconnects it. My (not very elegant) solution was to try every 15 seconds from the RPi to reconnect:
1. go to libreelec bluetooth settings and (re)connect the remote
2. ssh to the RPi
3. issue command echo|bluetoothctl , write down the MAC address of "Amazon Fire TV Remote", for example 11:22:33:44:55:66 (yours will of course differ!)
4. create cronjobs to try to connect every 15 seconds:
crontab -e, enter in editor:
* * * * * sleep 00;echo 'connect 11:22:33:44:55:66'|bluetoothctl > /dev/null 2>&1
* * * * * sleep 15;echo 'connect 11:22:33:44:55:66'|bluetoothctl > /dev/null 2>&1
* * * * * sleep 30;echo 'connect 11:22:33:44:55:66'|bluetoothctl > /dev/null 2>&1
* * * * * sleep 45;echo 'connect 11:22:33:44:55:66'|bluetoothctl > /dev/null 2>&1
save and quit editor
5. Press the "Home" button on the remote for at least 16 seconds
6. Enjoy reconnected remote
Exercise left for the impatient: more cronjobs, less sleep