In line with our transport options and for things to do, our suggestion would be to stay in Invercargill.
There are many hotel and motel options available depending on your budget. Booking.com would be your best bet.
If a house is better suited to your needs, Airbnb & bookabach both offer good options.
If you’re inclined to get a campervan you could consider Lorneville Holiday Park or Invercargill Top Ten Holiday Park.