There were enough people who had them installed when they bought the car new that there should be some used cars on the market with one already. The local VW Australia towbar made by Best Bars is the option to look for.
Surprised the VW dealer can't locate you one. As a price guide, fully fitted you should be looking around $1500-1600 for the VW option for the Mk6 including dedicated trailer control module electrics. Aftermarket from Hayman Reese or similar with bypass electrics will be around $800-1000.
Retrofitting yourself is possible but a little bit of a pain. When I had my Mk6 GTI I searched Gumtree, Facebook classifieds and wreckers to get the Best Bars towbar to fit on to mine.
I spent around $300 for one including the VW trailer control module. The physical towbar installation was easy enough, the electrics were a little more painful due to needing to make the CANBUS connection up near the OBDII port above the accelerator. The Mk6's come pre-wired in Australia from rear left of car where the trailer control module is installed to that wiring I mentioned above.
Pic is when I was fitting my bar, good fun!
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