G'day Mr Beach,
I know the area around Lake Douglas, Fox, Johnny, Nameless, etc is technically outside the national park. However, as shown by the map in
this post, it is part of the World Heritage Area. I suspect our friends at PWS would have a fit at the thought of a 20+ group with pit toilet, etc.
Ironically, I suspect Forestry Tasmania are your friends in this case. They manage their land for 'multiple-use' which means they allow higher impact activities in most forest reserves such as digging toilet pits, large groups, campfires (at certain times of the year of course), trail bikes, 4WDs, mining, etc... (Perhaps I was getting a little synical at the end there!)
More seriously, many of the forest reserves around the place do offer some very high quality camping opportunities in spectacular wilderness areas. Much of the Tarkine is in this category and there are numerous opportunities in that area that would be beyond the reach of the 4WD and trail bike fraternities. Also in the NW, I used to belittle the Dial Range as a bushwalking destination but I've since taken a liking to some of the secluded valleys and their myrtle forests. There are a few spots in there but care needs to be taken to pick which river flats are private property and which ones are ok for camping.
Mt Maurice in the North East could work. The route described by the Abels is a pleasant walk along an old logging road which ends at an old car park on the fringe of a myrtle forrest. A large camp there could be a base for exploration out onto the Maurice High Plains. I'm not too familiar with the NE but I expect there are several other similar forest reserves that could provide such opportunities. Mt Victoria is another place where there is a very large forest reserve.
As TasAdam said, the upper Mersey has a few opportunities. I'm guessing the spot he is talking about is the Mersey White Water Forest reserve between Lakes Rowallan and Parangana. It's right next to the road so it could be a bit dodgy.
The west coast also has some massive forest reserves and regional reserves (e.g. Mt Dundas RR) which could have some bushwalking opportunities. Perhaps the Tyndalls, utilising the 4WD track to Lake Spicer (not sure if this is still being used by 4WDs much any more).
Have fun! Are these early preparations for a BWT 2nd anniversary celebration in March?