Wordpress - WP REST API -- How to change HTTP Response status code?

You can return a WP_Error object in which you define the status code. Here's a snippet from the REST API documentation:

function my_awesome_func( $data ) {
    $posts = get_posts( array(
        'author' => $data['id'],
    ) );

    if ( empty( $posts ) ) {
        return new WP_Error( 'awesome_no_author', 'Invalid author', array( 'status' => 404 ) );
    }

    return $posts[0]->post_title;
}

In your case you could do something like:

return new WP_Error( 'page_does_not_exist', __('The page you are looking for does not exist'), array( 'status' => 404 ) );

The simplest way is to do this:

return new WP_REST_Response(null, 404);

Note: this is still a JSON response containing null as the body. You'd have to hook into the rest output filters, and change it to text/plain or something to output nothing.