They fly to GRU, and availability is very good - I checked the other day for flights in Feb and saw that you could still get EY J seats for most days in August.
EY are also on the lower point table compared to long-haul NZ, but it further. If I'm reading the tables correctly, MEL-AKL-EZE is 143.8k for J on NZ, and MEL-AUH-GRU is 192k for J in EY (one way) - but I'm obviously reading it wrong since I saw them for 125k to GRU this morning.
Are you actually going to EZE or is that where they fly to and you're going on to somewhere else?