Flying OUT of Sydney in business you would have had access to either (or both) of the AirNZ and Singapore lounges. In my experience they are very good about telling you about that at check-in.

However the OP is asking about showers on the way INTO Sydney, not out.

As has been said above, there are "public" showers in the international terminal - BYO everything. Otherwise the only option as you've said is to either fly business class on either Qantas or Virgin, or pay for the one-time pass with Virgin (max 2 hours). Both QF and VA have showers in their Sydney lounges.

Some time back someone did claim the new-ish Rydges hotel at the International terminal was offering paid access to their gym/showers, but I don't know that it was ever confirmed. A call to them would probably get to the facts.