Hi,

You should insert this code either into the functions.php file of the child theme or by using the CodeSnippet plugin available here: https://wordpress.org/plugins/code-snippets/

add_filter( 'register_post_type_args', 'soledad_template_post_type_args', 10, 2 );
function soledad_template_post_type_args( $args, $post_type ) {
    if ( 'custom-post-template' === $post_type ) {
        $args['show_in_rest'] = true;
        // Optionally customize the rest_base or rest_controller_class
        $args['rest_base']             = 'custom-post-template';
        $args['rest_controller_class'] = 'WP_REST_Posts_Controller';
    }
    return $args;
}

Now you can get the post template data via this URL: https://yoursite.com/wp-json/wp/v2/custom-post-template/

Regards,
PenciDesign.