My solution is specific to using Contact Form 7 and WP-SES.
In the contact-form-7/includes/ folder, I modified line 551 of classes.php.
Replaced
$headers = “From: $sender\n”;
with
$headers = “Reply-To: $sender\n”;
After this, all emails sent with Contact Form 7 will have the Reply-To set to the sender.
Keep in mind that any updates to Contact Form 7 will overwrite this change.