Ok, after a lot of googling and trial and error I solved it!
I generated a new key and put it in /config/.ssh/id_rsa
I copied the new key with ssh-copy-id to my LibreElec host
I also added a file /config/ssh_config with the following content:
Host *
StrictHostKeyChecking no
After that I was able to use the following shell_command without issues:
ssh -F /config/ssh_config -i /config/.ssh/id_rsa root@ip_of_libreelec 'reboot' 2> /config/command.log
The last part is just to see the output from the host when issuing the command.