Matt, bascially nothing going for road bike hirehttp://www.bikehiretasmania.com.au/
was run by Applebly's but they're out of that now.
Derwent bike hire – Cenotaph --- 0428899169 have a range of bikes but more your flat bar
Moto adventure – Macquarie Wharf – 0447556189 have electric and cruisers
Mona – Waterfront again cruiser style bikes
they're all about $20-$30 per day
There is also Art Bikes at Musuem or Arts TAS in Elisabeth Street, they're free and include some cool VanMoof bikes http://www.arts.tas.gov.au/artsatwork/artbikes
(Just read page Museum depot is closed during the renovations but you can get them from henry Jones Art Hotel)
They were intended to ride from gallery to gallery or arts location but people use them for general sight seeing.
You may have to forgo the challenge of Wellington and head out to MONA instead for some kulcha.
If you check any of the hire places out and have any info. to add it would be good to know.