It only took two and a half months, but I've finally gained access to the Perth Linux Users Group ABN. We had run into this problem when a long time committee member who had handled this side of things died late last year.
We realised that no one knew who else had access, so had to resort to printing out and posting paper forms to the ATO requesting that they update our records. It took three attempts before I had the correct documentation for them to accept the forms. What eventually worked was minutes from a committee meeting that included a motion appointing me as a "public officer". AGM minutes showing that I had been elected to the committee were not sufficient.
I guess the main lesson here is that if you're running a club or association, make sure you know who is listed as an associate and that there are multiple active members on that list. While you can recover via the paper forms, it's going to be far quicker for an existing associate to update the records.