If AMEX makes your payment due on a Sunday, they expect the payment on a Sunday. I incorrectly assumed that the next business day would be acceptable.
I also pay on-line via my bank and they will not process payments on Sunday. However, this does not matter to AMEX. Been a customer for 10 years and they charged me $32 in interest for making my payment on the next business day, which in my experience is standard practice. So a $32 lesson with "no way" to waive it according to the customer representative.
Once I am eligible to cash in my rewards, I will and close the account. Many businesses are starting not to accept AMEX anyway.
Product or Service Mentioned: American Express Credit Card.