Mick, I have a TopHat and know NSWSailor and Bluemoon very well. One of my reasons for buying our Top Hat 10 years ago was because of two boats in particular. "Caprice", in which Ian and Jan Mitchell circumnavigated the world over four years from 1974 to 1977
www.writingsfromjanmitchell.com/two-in-a-top-hat.html and their son Jamie's adventures in "Possibilities". Jamie traveled the world for 10 years in a Top Hat, arriving home in Sydney in 2009.
There are plenty of Top Hats for sale at very good prices, they have a lot of storage, but are quite slow when weighed down with supplies. If you're going non-stop, you'll need plenty of food and water, not to mention fuel for emergency. 3 to 3.5 knots average will be hard to keep up if you want to keep to a deadline.
There has been at least one Top Hat go around Australia too, if you drop over to the Tophatyachts.com website and search for Sundowner, you can read all about Rob's trip around anti-clockwise with plenty of stops to see the scenery. It also outlines some of his hairier moments on the WA coast, and in the Bight. It wasn't all plain sailing (sorry for the pun

) Sector Steve, Blue Moon and NSWSailor have all been to the Whitsundays in Top Hats, some more than once. So they're eminently capable, as are Compass 28 and a few others from the same era and style.
The question is, are you up to it? As others have said, there is no rush, simply sailing around is going to be very tiring, very boring and you won't see anything of the coastal areas you're passing. You'll need to keep plenty of sea room and keep out of the shipping lanes to get any sleep (witness Jessica Watson running into a container ship off QLD while she was asleep
www.yachtingworld.com/news/jessica-watson-hits-ship-11579)
I don't think you need do any courses, unless you really don't have any clue about sailing and ocean safety. Most of it can be read and studied on-line, then you can do as others have done and simply bite the bullet on a good solid starter boat. Others will have a different opinion, some will recommend to join a race boat crew to get experience.
I'm in Pittwater, and would be happy to show you over our old girl. She's not as flash as Phil's or Shaun's after their re-fits, but you'll be able to see what they're like to sail and how they can be set up for a trip such as yours.
If it was me, I'd be going and stopping at every port available and seeing whether there was a young lady up for an adventure, who wants to go with you.....

PM me if you want to investigate, and good luck getting it sorted!