Queenstown or Christchurch

I dispute that ther is nothing to do in Chch! Apart from driving around there are
1. Botanic Gardens -looking lovely at present & will be ramping up for the big flower show end of Feb;
2. Punting in the Park (or hire a xanoe & DIY);
3. Canterbury Museum, esp the Antarctic Section; also Quake City in Cashel Mall
4. Antarctic Centre;
5. Willowbank Wildlife Park (kiwi & kea esp) + Te Kotane Maori Culture Show
6. Transitional ("cardboard") Cathedral
7. Restart Container Mall, until April when rebuilding starts
8. Port Hills Gondola - fabulous views from the summit of an extinct volcano
9. Historic Tram
10. The many "gap-filler" art works including the Pallet Pavilion.
Possibly the reason you couldn't find the ticket you want is that there are no flights direct from ZQN to MEL. In a week a good choice would be to book flights LAX-AKL-CHC-MEL and then add a domestic flight to ZQN, stay 3-4 days and coach back to Chch, possibly including a night at Mt Cook enroute.
Or book open jaw: LAX-AKL-ZQN and CHC-MEL.

We were able to book a multi-stop on Air New Zealand from Los Angeles, Auckland, Queenstown, to Melbourne on Oct 1.