I have been making ATO payments for years using the Qantas American Express card. You pay the CC fee, but it's money I had to pay in any event. this is the website I use, and although it says optussmartpay.com I don't know, but could be a service used by the ATO (?) as the transactions all go through without a problem.
ATO - Card Payment
this is now on my QF account as a result of paying my personal tax:
Date Description Qantas Points Status Credits
Wed 11 Apr 2018 QANTAS AMEX PREMIUM CARD 36,340 0
AMEX will give .5 points per dollar spent with the ATO.
Considering the 1.45% cc fee, it's expensive but as mentioned above money I had to spend in any event.