Ryan
"technically" you have 3months to change your rego over to Tasmania plates BUT they have to prove that you have been here longer than 3months and its your permanent address.
There are a number of people that have gone 6-12months before being picked up for it so i wouldn't stress too much about that.
If you were to change it over then you would have to your car AIS (inspected) and passed off as road worthy (wont happy with the screamer) and then pay for your rego and plates, its a pretty simply process and even easier if you find someone that can look after you with the inspection
Police are no where near as harsh as Vic or other states, if you are not doing anything to draw attention to yourself or being a "hoon" you shouldn't have any problems at all.
My car runs external gate/screamer and was a daily (hasn't been on the road for a while ) and never had any real drama's with it...
As for track days there is always something happening on one of the tracks here: Symmons is about 15mins from Launie and Baskerville is 20mins from Hobart CBD.
In regards to tuning there are peeps that can help you in the north or south so i wouldn't rush too much to get it finished.
There will be another SAU cruise in the coming months so hopefully you can make it along and catch up with the Tassie members
Cheers GMB