Like you I been doing extensive research regarding your 146 memory .
I've looked back through the ANZ Facebook site as well as Ansett , RNZAF pro boards , aviation photo sites and more.
I cant find anything referencing an NZMC landing.
Ansett did fly one into Wanaka for the 1996 and 1998 War Birds air show.
If it had been parked close to the terminal, and photographed from the same angle as my screen shot ...the back drop would look similar.
I have found a photo of it parked ay Wanaka (static display area) , and an old video of it departing.
Here's the video link ;
https://www.youtube.com/watch?v=Km02c0Lx3HMI don't have permission yet to download the photo, but search for "BAe 146 at Wanaka" at Airliners.net
But still no mention of Mount Cook. I did discover information about one landing at Te Anau. Operated by Air National for Vincent Aviation with 18 pax onboard.