I’m 5 days into my South Island trip and I can say without a doubt - rent a car! It allows you so much freedom; I can’t imagine having to follow the schedule of a bus. And after hiking, you can just throw all your gear into the car and deal with it back at your hotel instead of having to manage everything in the cramp space of a bus.
Also, it seems like many of the day hikes I’ve been on, there are no buses that will take you there. Buses only seem to go to the big sites, like Milford Sound. I didn’t see any buses at the car park for Lake Marian Track or Routeburn Track, for example. If I’m wrong about this, someone please correct me.