Reconfigure PHPMailer so that WordPress AWS SES to send mail via Amazon. Change Username, Password, and Host to match your AWS settings. Then add to `mu-plugins` to use.

send-smtp-email.php

<?php
/***
Plugin Name: Amazon SES SMTP
Plugin URI: https://gist.github.com/onecooltaco/1c3e2d108fa2cd5c3d029987fa9b814c
Description: Send mail from Amazon SES
Version: 1.0.0
Author: Jeremy Leggat
***/

/*
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License, version 2, as
published by the Free Software Foundation.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA
*/

// Amazon SES SMTP to send PHP mail.
add_action( 'phpmailer_init', 'amazon_ses_smtp' );

function amazon_ses_smtp( $phpmailer ) {
    $phpmailer->isSMTP();
    $phpmailer->SMTPAuth = true;
    $phpmailer->Username = '####################';
    $phpmailer->Password = '#################################';
    $phpmailer->Host = '{email-smtp.us-east-1.amazonaws.com';
    $phpmailer->Port = 587;
    $phpmailer->SMTPSecure = 'tls';
}