Loading specific bootstrap mode in hook_menu
It's not possible, because hook_menu() is invoked when Drupal is completely bootstrapped, also because only when Drupal has been completely bootstrapped, all the modules are loaded; only in this case all the menu callbacks defined in hook_menu() are available.

If you really need to limit the bootstrap to a phase, then you can write a file that is similar to index.php, for example.

define('DRUPAL_ROOT', getcwd());

require_once DRUPAL_ROOT . '/includes/bootstrap.inc';
drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL);
menu_execute_active_handler();


An example of file used for a specific task is cron.php.

define('DRUPAL_ROOT', getcwd());

include_once DRUPAL_ROOT . '/includes/bootstrap.inc';
drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL);

if (!isset($_GET['cron_key']) || variable_get('cron_key', 'drupal') != $_GET['cron_key']) {
  watchdog('cron', 'Cron could not run because an invalid key was used.', array(), WATCHDOG_NOTICE);
  drupal_access_denied();
}
elseif (variable_get('maintenance_mode', 0)) {
  watchdog('cron', 'Cron could not run because the site is in maintenance mode.', array(), WATCHDOG_NOTICE);
  drupal_access_denied();
}
else {
  drupal_cron_run();
}


Generally, there isn't the need to limit the bootstrap, only to serve data in JSON format. Creating a file like cron.php should be done for a dedicated task that is time consuming, or for limiting the pages being served, as with install.php.