it looks like mod/turnitintool/index.php is hard-coded to write the $SITE->fullname as the header

/// Print the header
turnitintool_header(NULL,$course,$_SERVER["REQUEST_URI"],$strturnitintools, $SITE->fullname, $navigation, '', '', true, '', '');

Could this be replaced with
/// Print the header
turnitintool_header(NULL,$course,$_SERVER["REQUEST_URI"],$strturnitintools, $course->fullname, $navigation, '', '', true, '', '');