If it was sent via SES, it never went out via Exim. It would have connected via SMTP to SES to send, which would never trigger Exim.
By default, unless you enable some sort of logging, there would not be anything available from AWS. I personally use SNS topics to track delivery of emails. I pipe them to SESDashboard track success/fail, spam complaints, opens, clicks, etc. It allows me to do exactly what you're trying to do... prove what happened. For example, in this case, I can see not only was the message successfully sent, but it was received by the destination mail server AND that the recipient opened the message... then actually clicked a link inside of that message.
Clicking on the event log would show more detailed info about the event itself.