It's not really the relevant space for them. Not that the US doesn't want anything to do with them. Its not five eyes, its not the UN, its not NATO.
From Australia's side, Australia would be more than happy to entertain pretty much *ANY* defence proposal from NZ. If NZ wanted to co-man submarines, operate F-35's, joint construction of vehicles or naval units. Joint forward basing etc. It would be hard to for see except for very specific multinational agreements like AUKUS projects that would exclude any national other than those specifically involved (like nuclear reactor technology).
Australia would expect strong support for sensible regional action by NZ. Like Timor. We would certainly expect to jointly coordinate responses and not act independently, even in HDAR or peace operations. Again at least some coordination of exercises and practice of integration.
But its completely up to NZ. Australia isn't fending off any NZ proposal.
Australia on the other hand has been pushing for a long time. We wanted in on nuclear submarines, Nuclear weapons, F-35, F-22, F-111, BMD, hypersonics, AI, sub combat systems, torpedo development, US marine basing, US base investment. Some of these are knocked back. The US and the UK subverted Australia gaining its own independent nuclear weapon capability (for good reasons). F-22 was also refused (for good reasons), other items are greenlit after lots of wrangling.
The relationship AU has with NZ is different. I can't think of a single case of NZ wanting in on something defence related and AU saying, no.