I fly either that route (as part of a GRU-SCL-AKL-MEL) or the GRU-SCL-SYD-MEL route about 8+ times a year, though typically in business.
There are really only two airlines to get you from SAO to AKL, and they're Qantas and LAN, who each code-share the other's flights.
LAN's fares are usually cheaper tha Qantas, even for exactly the same flight (ie a LAN-marketed-fare on a Qantas-operated flight). The service and amenities in business at least are almost identical on both the LAN and Qantas flights out of Santiago (SCL). In economy I suspect Qantas will be better, as I don't think LAN have seat-back entertainment in economy.
Because of the lack of competition on these routes, you will not find great variance between days of the week, especially if booking from further out. As the date gets closer, the weekends and Monday and Friday tend to book up, leaving only the higher fare classes aqvailable, so 1-4 weeks out you may see perhaps as much as a 20-30% difference in fare between a mid-week flight and the other days. But this depends entirely on what has filled up.
I recommend you book a LAN-marketed flight on Qantas metal, via an OTA such as Expedia. This is likely to be the best value.