Node custom property exposed as extra field
Looking at Entity module's hook_field_extra_fields() implementation:

/**
 * Implements hook_field_extra_fields().
 */
function entity_field_extra_fields() {
  // Invoke specified controllers for entity types provided by the CRUD API.
  $items = array();
  foreach (entity_crud_get_info() as $type =&gt; $info) { ... }
  return $items;
}


It only looks for entity types provided by the CRUD API -- in other words, entities whose controllers inherit from EntityAPIControllerInterface. Node entities use NodeController.

You can still do this conventionally, outside of Entity API -- implement hook_node_view() to insert your pseudo-field in $node['content'], then implement hook_field_extra_fields() to expose that pseudo-field to the node's "Manage display" tab.