Wow that's great. No Amex here in Brisbane and we do pay a surchargeMy rates are with Sydney City Council.
They have an online portal to pay your rates where they accept AMEX with no surcharge.
Can't schedule when to pay but I just set a calendar reminder a day after my new statement starts and pay. Helps me earn 50% more VA points or 11% more SQ than my CB Signature card.
Not sure what other cards you have but you might be able toi earn a few extra points this way.