Thanks! It was so obvious I had completely missed it, it didn’t occur to me I also had to change the SMTP host.
For the record, AWS SES SMTP host is email-smtp.us-east-1.amazonaws.com if the SES is configured in us-east-1.
Source: https://docs.aws.amazon.com/ses/latest/DeveloperGuide/send-an-email-using-smtp.html