Your backup's "data" does not match your backup's "system". It's unclear to me what's on the "Big Sur Backup", but it seems like you installed updates to the OS and then didn't copy them to the backup.
You can install a current version of Big Sur over the backup to bring it up to date (assuming the data isn't from a later OS, like Ventura) and then it should boot properly.
But it seems it would be more sensible to install the 13.1 update, where Apple has fixed the notification issue.
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--Dave Nanian
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