How do I add to the 'allow in background' list?

It is usually up to the app to prompt you to give permission for the app to add itself to the list. I'm not sure how you would go about manually adding it (if that is even possible) since it may be a helper app that gets added & not the main app.

Check with the developer of the app for assistance if their app is not showing in the "Background" section where you can toggle it on/off.

Should that backup software be Carbon Copy Cloner ?

https://support.bombich.com/hc/en-us/articles/20686443528727-Granting-Full-Disk-Access-to-CCC-and-its-helper-tool

Otherwise, you would need to contact the Developer of this unknown backup software Developer