WPvivd Backup Pro allows you to separately back up WordPress core files.
You can choose whether to back up WordPress core files or not when making a manual backup or a setting up backup schedule.
Note: WPvivid Backup plugins(Free and Pro) do not restore the .htaccess and wpconfig.php files as it can easily cause problems to restore the same .htaccess and wpconfig.php files on a different site. You should download and copy them manually as needed.