I don't see how you have any choice, of course you have to grant Carbonite full disk access, how else can it back up all your files to the cloud?
Carbonite will have possession (and safekeeping) of all your files from your Mac in its cloud backup storage areas if you use its backup service. If you don't trust them, uninstall their app and don't use it. I think they are a trustworthy company.
With Monterey, you open System Preferences, go to Privacy and Security settings, and scroll down to find the Full Disk access options.