pdwtrip, I have good news!
As a silver Exec Club member, you are oneworld sapphire (blue oval on card). Sapphire card holders ARE indeed eligible for entry to the domestic Qantas Club.
(The exception you mention relates ONLY to QF Gold members accessing the domestic Business lounge, and is basically an historical remnant from pre-oneworld days).
As a oneworld sapphire, you are also entitled to bring in one guest, so your wife is fine as well. Simply show your BA silver Exec Club card and your onward boarding passes, and you will certainly be admitted.
See here for full details of Qantas Club entry (scroll right down to the bottom for the oneworld status access):
So this will get you access to the SYD domestic Qantas Club. When you exit through Immigration at SYD international, turn right and keep going to the QF domestic transfer area. They will then take you, by bus across the tarmac, to the domestic terminal. Then go upstairs and enter the Qantas Club. They have showers etc there and it is quite pleasant.
One other point: you don’t mention what class you are booked in for your domestic flight SYD-MEL. If it is in Business class, then you can also use the Qantas domestic Business lounge.
When you depart Australia, if you are flying F out of SYD, you will be able to use the exceptional Qantas First lounge. Make sure you get to the airport quite early as this is a wonderful experience (and do try to book a spa treatment; best booked at least 24 hours in advance).
As to the issue of arrivals lounges – people from the UK are very fortunate – Heathrow has dedicated arrivals lounges for quite a number of airlines. Understandably, UK residents then often expect that airports in other major cities will have them too. The truth is that VERY few airports anywhere in the world, other than LHR, have arrivals lounges. LHR would have more than the rest of the world put together, I think.
Hope that clarifies everything. Enjoy your trip!