I’m not sure why you have an aversion to doing this correctly – either on the server a la
PrintMotd no
PrintLastLog no
and
#/etc/pam.d/ssh
# Print the message of the day upon successful login.
# session optional pam_motd.so
Or adding ~/.hushlogin for each user.
Hint, for ~/.hushlogin, add it to /etc/skel so new user home directories are created with the file.
Update:
Without more information about your backup cron job, my only other suggestion is to redirect the output of the command to a file (or let cron capture it in email) and the output of the ssh session to /dev/null. Something like:
0 0 * * * ssh backuphost "backup_script_that_writes_to_a_log" >/dev/null
Or
0 0 * * * ssh backuphost "backup_command 2>&1" >/dev/null
I’d have to play around with the commands a bit, but that should get you started.