Yep, the CPRID hack via SIC is your only remaining option. You'll have to adapt sk106490 for the commands you need. This should get you very close:
Assuming you get a clean SSH configuration uploaded via Gaia API put-file, you can move it into place (assuming you uploaded the file to /home/admin):
$CPDIR/bin/cprid_util -server <IP_address_of_Security_Gateway> -verbose rexec -rcmd "mv /home/admin/sshd_config /etc/ssh"
Restart SSHD (Gaia still uses the SysV style scripts, not systemd):
$CPDIR/bin/cprid_util -server <IP_address_of_Security_Gateway> -verbose rexec -rcmd "service sshd restart"
Gaia API is on by default and open to the 'admin' user. You can use the Ansible module for it, or craft your own method manually. You can't use SCP or SFTP because SSH is broken.
(Edit: typo)