The plan is.
Day 1- Arrive in CHC morning. Go around and see CHC then catch a Crusaders game at night.
Day 2-Pickup Rental car at airport. Drive. Planned a detour and stopover at Mt Cook National Park, just to take in the sights and some picture taking. Start moving maybe a short stop at Wanaka. I recall it being somewhat a nice skitown so again, just to take in the sights. Drive to Te Anau. Sleep in Te Anau.
Day 3-Leave for Milford Sound in the morning, planned to get the 1st Jucy Cruise. Leave for Invercargill, take in the sights a long the way. Mirror Lake, Lake Te Anau etc. Short layover at Invercargill. Based on my reading not much in Invercargill so again, just take in the sights. maybe take a short detour to Bluff if I have the time. Drive to Dunedin, loiter and sleep in Dunedin.
Day 4-Drive to Queenstown. Loiter and sightseeing around Queenstown. Sleep in Queenstown.
Day 5-Return car in the morning. Loiter around Queenstown. Flight back to Auckland at 4pm.
It was a last minute plan to go to South Island. Didn't factor in a trip to South Island initially, thought of just going North but changed last minute so decided to just rent a car and drive around. For the record, I have done an Auckland-Milford Sound-Auckland trip driving before in 2004 so this is technically revisiting.
Google Maps states it's a 1600+km drive in total, without factoring the 30km detour should I decide to take it to Bluff. So I'm down to about 700km or so per day in those 2 major days of driving. One of my possible variable is skipping Mount Cook National Park. Saves me about 2 hours or so of driving. Bluff is another variable.
I don't think there's enough time especially Milford Sound - Invercargill. Your itinerary too pack IMHO