Generally speaking most of the Conservation Areas and Regional Reserves are off-limits to dogs/cats/domestic pets for the same good reasons as they are for National Parks. The exceptions to that rule are listed here:
https://parks.tas.gov.au/explore-our-pa ... s-in-parks - sounds like you've already checked it out though

Certainly, none of the 'usual' multiday walking routes qualify.
You could probably string together a good multiday coastal route in either the Arthur-Pieman or Waterhouse Conservation Areas. Though I'd question the appropriateness of taking a dog through beaches/dunes and potential seabird breeding areas, even if it isn't specifically against the rules.