Confirm no NZ ff# in the booking, and LH ff# was used?
Was the issue with carry on only, or also checked baggage?

I suspect it was the fact the domestic sector was under a SQ code-share - so the sector XXX-AKL was a SQ flight operated by AirNZ.
The website is unclear if the benefits for 14kg carry require you to be on NZ marketed and operated flights or not...... But as noted above by nzkarit , there is no Star Alliance carry-on benefit for status holders.

In May, despite being NZ Elite and on NZ metal on native flight number, I was not allowed in the Lounge due to having my TG Silver credentials in the booking as I wanted to regain TG Gold.

This is a reduction in benefits that would usually be a provided across the Star Alliance - I should be able to access any Star Alliance lounge (including NZ's lounges!) by showing my star Gold card/credentials (even NZ's) - but AirNZ insists that the ff# of the program where you hold status (only) is in the booking.