Hi,
Not sure if this is possible. Until now I didn't saw this. there are multiple ways in manually making backup's and upload them via SFTP / FTP in single commands. There is also option to restore these backups without reboot. In order to achieve similar process like your I see a lot of organizations enable the following snmp trap:
snmp-server enable traps startup-config-change
when the management system than receives this trap it automatically makes backup. Maybe this will also work for you.
Regard, Dobias