Set up a custom WordPress REST API controller class for the Page post type.

custom-wordpress-page-rest-controller.php

<?php

/**
 * Set up a custom REST API controller class for the Page post type.
 *
 * @author Kellen Mace
 *
 * @param  array  $args The post type arguments.
 * @param  string $name The name of the post type.
 *
 * @return array $args The post type arguments, possibly modified.
 */
function km_set_custom_page_rest_controller( $args, $name ) {
	// If this post type's name is not 'page', do not modify its args.
	if ( 'page' !== $name ) {
		return $args;
	}

	// Tell WordPress to use our KM_REST_Pages_Controller class
	// for Page REST API requests.
	$args['rest_controller_class'] = 'KM_REST_Pages_Controller';

	return $args;
}
add_filter( 'register_post_type_args', 'km_set_custom_page_rest_controller', 10, 2 );