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Peter, I would expect it is doable but watch out for the paperwork/regulatory issues. If your boat is not out of Florida within 90 days of purchase, you'll need to pay Florida sales tax, 6-8% depending on county. One way to drag that out might be to work with a flexible seller, pay them eveyrthing except the final dollar and then "conclude" the sale after you've had a chance to work. There are hazards in that too, so simply planning to be gone in 89 days might be simplest.
There's also the matter of flagging the vessel, presumably you would want to put an Aussie flag on it, once that's done you have a timeclock to get it out of the US or apply to get it back in, or remain in. Regulations here often change on January 1st for the new year, so find out what you'll need to do now, but check again before the new year as well.
The choice of boats is a wide one, but that's a whole topic in itself, see the many threads about what is a bluewater boat, and take a look at what other ARCs have been sailed with, with comments from or about them. With a trip like that you may very well decide that unless the boat comes with new sails including storm sails, that you want a new main, new genoa, new job, new storm sails, and kaching that could all add up to a fast $20-25,000 for a large boat in prime season.
I would expect there will still be many boats on the market, but between finding a boat, making an offer, negotiating, booking a survey, seatrial....I'd be surprised if you could locate and take possession of "the right boat" in under a month. So do give yourself time up front, especially in Florida where everything seems to take a week longer than you'd expect.