Brisbane city gets points on Citi cards..
My 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.