I had HD200 and then I ugraded to Piaggio Beverly. I found 350cc was plenty enough for out of town trips, I think it would serve you well on Tassie roads. It has large wheels too, which is nice for stability. I don't think you really need BMW or Burgman - it's not so much the engine size that is the problem, but their bulk, and in case of the beemer, also the height - it's a very tall bike! Beverly sits in a nice spot , good balance between ability and size.
If Piaggio is not a good option in Tasmania in terms of dealer support, I would suggest looking at other scoots in that size. Kymco Downtown for example, also 300cc, or even Suzuki Burgman 200cc - going by write-ups here on our site it coped with open roads surprisingly well.