Define basic custom field and callback function

setup-meta-transient-rest-api.php

<?php

/***
Register custom field for existing endpoint with callback function
Endpoint: /wp/v2/your_post_type

Note: Could also be a custom endpoint using register_rest_route
***/

register_rest_field('your_post_type', 'custom_field_name', array(
    'get_callback' => 'yournamespace_endpoint_callback',
    'schema' => null,
));

/***
Callback function - if you use a custom endpoint you may have to change how you get $post_id
***/

function yournamespace_endpoint_callback( $object ) {
  
  $post_id = $object['id'];

  return yournamespace_get_post_transient( $post_id, 'yournamespace_data_example', 'yournamespace_get_custom_meta' );

}