Couple of quick points - the Overland Track weather isn't much worse than anywhere else in Tassie can be. It's not that it is always bad, just that it can be, and if you're not ready for it you can be in a bit of bother.
The east coast / Freycinet is a good option and very nice... And the weather is
usually better...
You can walk in to the bottom of Wineglass Bay and camp there, or you can go further up & over Mt Graham and down to Cooks Beach. Or you can get to Cooks Beach via the eastern side, a close look at a map will show you where I mean. You can do a circuit - down to Wineglass Beach, then over Mt Graham to Cooks beach, then back along the coast / Hazards Beach to the carpark.
Only a short distance in, but Shadow Lake is very nice, and from here you can climb Little Hugel without too much bother. Or a day walk around to Mt Rufus from Cynthia Bay (Lake St Clair) is pleasant.
Or take the early ferry across Lake St Clair to Narcissus, then up to Pine Valley and from here you can either take a day trip up to the Labyrinth if the weather is good (pretty exposed up there), or maybe climb the Acropolis. And you could walk back to Echo Point from Pine Valley in a comfortable day, and another comfortable day back to Cynthia Bay, taking your time appreciating the forest along the shores of the lake.
No doubt you will get other suggestions.
If you start
HERE you can zoom in, drag around, and see a lot of places.