Luke, the best time of year is now. The boat in the link above is Enchanter, Lance is the skipper and he is the best man for the job. Your trip is very weather dependent, and there is every chance you will be tied up to the wharf for your week there. The Greymouth bar will scare the crap out of you, its one of the worst in NZ. You ask about when the bigger fish are there? You want the smaller fish if you are entertaining any kind of shot at landing one on popper or jig tackle. Mogi got spooled once, and landed his second fish, est. 250kg, and thats a smaller NBFT.
Personally I have not traveled down to chase these monsters, as the actual hookup of the tuna requires no angler skill, but if a like minded crew was keen to try on poppers or jigs, I'm keen as. But I think it would be the SBFT would be the better targets. They are only about 150kg...