You might also see what happens if you create an alias file (aka link file) from index.php to drupal/index.php... does cPanel allow creation of link/alias files? I'm note sure...
If you try this, be sure to have all your Drupal config file settings matching the full domain (www.example.com/drupal), because it'll need to pick that up to run properly.
What you are trying to do in theory basically (if I understand everything correctly) is make the page: www.example.com/drupal/index.php show up as www.example.com/index.php . The $baseurl is a setting that tells drupal
how
to formulate it's path. This does not change the server's behavior. Now,
you
could setup a subdomain. E.g. drupal.example.com, which may help.
Hope all that made some sense. Anyone correct me if I'm wrong. I'm not too familiar with cpanel, so perhaps there is a setting he/she could tweak?