If you want to ride to the summit I’d recommend the CX bike. Chuck a decent amount of pressure in your tyres, 'cause it’s a long climb up. It’s doable fixed, but the descent would be absolute hell. Pipeline Track (starts at Fern Tree) is a lot of fun on a CX bike too. It follows the contour and climbs very, very slightly as you head south, so it’s a fun ride out and a super fast smash back.
The only way to the summit is on the road. There are a few fire trails that’ll get you a bit of the way, and some trails that’ll get you half-way on an MTB, but CX or fixed will definitely be road for the most part.
If you’re in South Hobart (which is on the way to the mountain), stop in at Cyclingo (466 Macquarie St) and I can give you some directions if you’d like!