David,
Did this a year ago after my brother and his then fiance, now wife, moved back to her hometown just outside of Melbourne. We were able to play Kingston Heath, Yarra Yarra and Commonwealth. Royal Melbourne was closed to visitors at the time due to a severe problem on the greens.
{For those of you in that neighborhood, was this ever cleared up? It was a fertilizer mix-up I believe.} We played YY and Commonwealth with member friends but arranged Kingston Heath just as you suggest, a letter of introduction followed closely by a phone call. Call them, don't wait for them to call you. I also played NSW alone after my brother set up a tee time. I got the impression that was done just the same as KH. We also played Kooringal in Williamstown.
Other than the flies (sorry boys, I always give the bad medicine first
) Golf in Melbourne is special. In mid-December the greens were spectacular and I was given the impression it was a normal year. Well worth carving out some golf time. Some of the local boys will give all the details you need, but you have not mentioned what part of Australia this trip may be to.