As I said, and I am sorry that you don't understand this, there is no single back end for JQ and QF, especially when it comes to credit card transactions.

In this case davem4 is suggesting to roll back the JQ transaction on the card and in which case QF would be none the wiser. So the OP would get a letter from the bank saying JQ was disputing the information. Not from QF.

Based on the fact that the PAX had done the right thing and JQ failed to deliver part of the service the minimum would be a refund of the JQ fare for the PER-MEL flight which through no fault of the PAX has nothing to do from the PAX perspective with QF. If JQ won't provide this then the OP has every right to go back to AMEX, who I have found to always be excellent with chargebacks/ duplicate transactions/ disputed charges. But this issue has nothing to do with QF, they are just the operating carrier of a JQ codeshare. It is all about 3K/ JQ.