I would implement hook_node_access() to allow anonymous user to see the nodes on the front page (use the function drupal_is_front_page()):

function MYMODULE_node_access($node, $op, $account) {
  if ($op == 'view' && drupal_is_front_page() && $account->uid == 0)
     return NODE_ACCESS_ALLOW;
  }
  return NODE_ACCESS_IGNORE;
}

What kind of permission module are you using ? Is it possible you implement yourself these permission ?

I would restrict node access to anonymous user using a custom module implementing hook_node_access() and allow them to see the nodes on the front page (using the function drupal_is_front_page()):

function MYMODULE_node_access($node, $op, $account) {
  if ($op == 'view' && $account->uid == 0) {
    if (drupal_is_front_page()) {
      return NODE_ACCESS_ALLOW;
    } else {
      return NODE_ACCESS_DENY;
    }
  }
  return NODE_ACCESS_IGNORE;
}