Set error message to post meta.

set_notification_status.php

<?php function set_notification_status( $post_id, $post ) {
	if ( ! $post ) {
		return; // Exit if there is no response.
	}

	// @see https://developer.wordpress.org/reference/hooks/new_status_post-post_type/#comment-4335

	$author    = $post->post_author;
	$name      = get_the_author_meta( 'display_name', $author );
	$email     = get_the_author_meta( 'user_email', $author );
	$title     = $post->post_title;
	$permalink = get_permalink( $post_id );
	$to[]      = sprintf( '%s <%s>', $name, $email );
	$subject   = sprintf( 'Published: %s', $title );
	$message   = sprintf( 'Congratulations, %s! Your article "%s" has been published.' . "\n\n", $name, $title );
	$message  .= sprintf( 'View: %s', $permalink );
	$headers[] = '';
	$sent      = wp_mail( $to, $subject, $message, $headers );

	if ( $sent ) {
		$response = array(
			'code'    => 'success',
			'message' => 'Email notification(s) sent sucessfully',
		);
	} else {
		$response = array(
			'code'    => 'error',
			'message' => 'Failed to send email notification(s)',
		);
	}

	update_post_meta( $post_id, 'email_notification', wp_json_encode( $response ) );
}