With all of that in place, the SEs will see the same thing that the "public" sees.
Whether for you or others reading. A recommended process to accomplish this and maintain relatively uninterrupted operation is to first copy the files to this sub-directory. Include anything/everything that is to remain operational from your site such as the htaccess file, subfolders, etc...
Modify the new includes/configure.php (remember to alter the permissions to 644 before saving changes) and admin/includes/configure.php to reflect in this case (for now) that the site files and uri are in a sub-folder. (DIR_FS_ and DIR_WS_)
Attempt to access the site from the new sub-directory and verify basic operations work.
When satisfied, modify the configure.php files from above to remove the sub-directory modifications made to the constant(s) beginning with DIR_WS_
Make the change that your host does to redirect traffic to this sub-directory such that users will still see the site without a sub-directory.
Make a minor modification to one of the files such that an additional character or symbol appears on a specific page (not a database related file, but an actual file and do so through ftp rather than the admin, though suggest modifying one of the define pages found in: includes/languages/YOUR_LANGUAGE/html_includes/YOUR_TEMPLATE. This way you can verify via the admin that the admin is accessing the correct location).
While the change should occur rapidly, verify (after clearing cache/cookies) that when accessing the site, that the minor modification is visible. There may still be other servers with the host that need to propagate the change "internally", but you should now be able to initiate a rebuild/upgrade in yet another parallel sub-folder to be referenced possibly with a sub-domain.