WP-API restfull JSON and post meta?

Custom posts ignore the normal filter – have a look at class-wp-json-pages.php to see how to do build one for custom posts.

<?php
/**
 * Page post type handlers
 *
 * @package WordPress
 * @subpackage JSON API
 */

/**
 * Page post type handlers
 *
 * This class serves as a small addition on top of the basic post handlers to
 * add small functionality on top of the existing API.
 *
 * In addition, this class serves as a sample implementation of building on top
 * of the existing APIs for custom post types.
 *
 * @package WordPress
 * @subpackage JSON API
 */
class WP_JSON_Pages extends WP_JSON_CustomPostType {
    /**
     * Base route
     *
     * @var string
     */
    protected $base="/pages";

    /**
     * Post type
     *
     * @var string
     */
    protected $type="page";

    /**
     * Register the page-related routes
     *
     * @param array $routes Existing routes
     * @return array Modified routes
     */
    public function register_routes( $routes ) {
        $routes = parent::register_routes( $routes );
        $routes = parent::register_revision_routes( $routes );
        $routes = parent::register_comment_routes( $routes );

        // Add post-by-path routes
        $routes[ $this->base . '/(?P<path>.+)'] = array(
            array( array( $this, 'get_post_by_path' ),    WP_JSON_Server::READABLE ),
            array( array( $this, 'edit_post_by_path' ),   WP_JSON_Server::EDITABLE | WP_JSON_Server::ACCEPT_JSON ),
            array( array( $this, 'delete_post_by_path' ), WP_JSON_Server::DELETABLE ),
        );

        return $routes;
    }

    /**
     * Retrieve a page by path name
     *
     * @param string $path
     * @param string $context
     *
     * @return array|WP_Error
     */
    public function get_post_by_path( $path, $context="view" ) {
        $post = get_page_by_path( $path, ARRAY_A );

        if ( empty( $post ) ) {
            return new WP_Error( 'json_post_invalid_id', __( 'Invalid post ID.' ), array( 'status' => 404 ) );
        }

        return $this->get_post( $post['ID'], $context );
    }

    /**
     * Edit a page by path name
     *
     * @param $path
     * @param $data
     * @param array $_headers
     *
     * @return true|WP_Error
     */
    public function edit_post_by_path( $path, $data, $_headers = array() ) {
        $post = get_page_by_path( $path, ARRAY_A );

        if ( empty( $post ) ) {
            return new WP_Error( 'json_post_invalid_id', __( 'Invalid post ID.' ), array( 'status' => 404 ) );
        }

        return $this->edit_post( $post['ID'], $data, $_headers );
    }

    /**
     * Delete a page by path name
     *
     * @param $path
     * @param bool $force
     *
     * @return true|WP_Error
     */
    public function delete_post_by_path( $path, $force = false ) {
        $post = get_page_by_path( $path, ARRAY_A );

        if ( empty( $post ) ) {
            return new WP_Error( 'json_post_invalid_id', __( 'Invalid post ID.' ), array( 'status' => 404 ) );
        }

        return $this->delete_post( $post['ID'], $force );
    }

    /**
     * Prepare post data
     *
     * @param array $post The unprepared post data
     * @param string $context The context for the prepared post. (view|view-revision|edit|embed|single-parent)
     * @return array The prepared post data
     */
    protected function prepare_post( $post, $context="view" ) {
        $_post = parent::prepare_post( $post, $context );

        // Override entity meta keys with the correct links
        $_post['meta']['links']['self'] = json_url( $this->base . "https://wordpress.stackexchange.com/" . get_page_uri( $post['ID'] ) );

        if ( ! empty( $post['post_parent'] ) ) {
            $_post['meta']['links']['up'] = json_url( $this->base . "https://wordpress.stackexchange.com/" . get_page_uri( (int) $post['post_parent'] ) );
        }

        return apply_filters( 'json_prepare_page', $_post, $post, $context );
    }
}

Replace “Pages” with “MyCustomPostTypes” and page with “mycustomposttype”. Just be careful not to rename internal WordPress code that also uses the term page

Note: probably best to add this as a plugin rather than change the JSON-WP-API plugin

/**
 * Plugin Name: MyCustom JSON App API
 * Description: MyCustomPost handler for the JSON API
 * Dependency:  This plugin requires JSON-WP-API Plugin!!!! 
 * Author: 
 * Author URI: 
 * Version: 
 * Plugin URI: 
 */