I'm not familiar with offerings in your area. Perhaps it would be helpful to speak of more distant offerings, and perhaps others can say what's similar or available in your area?
From a U.S.-based perspective, if I wanted to dive with tiger sharks, I'd have 2 main options. Either go to southeast Florida, around the city of Jupiter, and head out with Emerald Dive Charters and watch a guy hand feed sharks (time of year's a big factor on species here), or else do probably a live-aboard trip to an area off Grand Bahama Island called Tiger Beach, where a 'scent triangle' or other means of baiting them in would be used. Cageless is offered.
For great white diving, it's either Guadalupe Island off the Pacific Coast of Mexico, or South Africa; those I hear mentioned most, and generally cage diving. It's my understanding GWS diving without a cage happens, but is more a minority activity.
Aside from GSWs being a tad bigger and a lot scarier looking, I'm not sure what cageless tiger diving seems so much more accepted...until I take close look at one of those massive open-mawed GWS close-ups, and think 'Noooooo.....'
Just for post completeness, if you like big, potentially dangerous sharks, there is the oceanic white-tip shark diving off Cat Island in the Bahamas.
Putting aside reef sharks drawn by feeding/chumming, people out of the U.S. have a couple of good options for 'medium' shark encounters in numbers; wreck diving with sand tiger sharks out of North Carolina, and the lemon shark migration/aggregation off southeast Florida. Would a bunch of 8 - 10 foot sharks be of interest? There's some place in Mexico where you can dive with bull sharks; people seem to regard them as much more potentially dangerous than sand tigers or lemons.
The above assumes you don't spear fish. You take that up, your shark encounter 'options' may change considerably, and someone else can talk about that.
Those of your familiar with Australia, are there local regional offerings analogous to the above? Is anyone aware of anywhere in the world where people routinely dive with a good probability of seeing tiger or GWS without using any form of baiting? Reliable enough you'd book a trip there just to see them?
Richard.
P.S.: You sure you don't want to get some more experience with medium sharks before you jump in with something 14 feet plus? Just a suggestion...