Yes, you do need the AIP
Having just sat the exam I can now say that you don't need the AIP to pass but there are a couple of questions where you will need to look up the answer in the AIP. Things like if you do the XXXX arrival from YYYY waypoint into NZAA then what entry will you do to enter the ZZZZ hold. If you fly into NZAA a lot then you may know but I certainly didn't. Another one was how long is the outbound of the WWWW hold (sorry, I just can't remember all the names).
So if you can handle having to guess a few questions then during your study just look online at aip.net.nz, otherwise I'd buy them (which is what I ended up doing). The one good thing about buying them is you get to see what parts of the GEN and ENR sections are actually in Vols 2 & 3 as it's actually much less than I envisaged. So if you counted on being able to look some stuff up in the exam rather than remembering it you may have to think again.
Anyway, hope that helps.