Efiled '23 return on 4/14/24. Rejected. Have notification that I had until 4/20 to mail paper copy. Got IRS penalty notice on 8/5. How can I prove that I mailed on time?

IRS says my return wasn't processed until May 13. They say they don't have a record of the postmark date. I didn't send certified mail so how do I prove that I mailed it on 4/16/24 within the grace period following a rejected efile.