AIR NZ FLT OPS Marty .... is, I think, probably the only definitive source of that sort of information.
Whether or not they'd assist with any such query, if approached, is an entirely different matter .... one really needs to know/have "personal access to someone within that department" to be able to access information such as that.
You say "October 1990", so, that automatically rules out any of the B763's .... since our first (ZK-NCE) wasn't delivered until June 1991.
Therefore it could only be a B762 .... either ZK-NBA, -NBB, -NBC, -NBD (acquired from VARIG during October 1987 .... and which operated with the VARIG stripe, "AIR NEW ZEALAND" titles, and a white tail for a couple of years prior to progressively receiving the full AIR NZ livery by around 1992/3), -NBE (the last factory fresh B767-219ER delivered to AIR NZ), -NBF (acquired from CHINA AIRLINES during December 1989),
"there was never any ZK-NBG", -NBH (also acquired from CHINA AIRLINES during December 1989), -NBI (acquired from BRITANNIA AIRWAYS during May 1990), or -NBJ (also acquired from BRITANNIA AIRWAYS during September 1990).
Bit of a wide range there Marty (I know) .... but .... less wide a range than might otherwise be the case without being able to exclude all of the B763's on the basis of the dates you mention

Again .... FLT OPS is probably the only way you could source that sort of definitive information .... short of someone else whom you may know down there and might have photographed that particular flight at CHC on either of the dates you mention and from which the/an aircraft might possibly then be positively ID'd
Mark C
AKL/NZ