Should an Indonesia, India , Taiwan or someone else come on board, NZ will no doubt be influenced.
It interesting to see who they are trying to pitch to.
I think NZ is much closer to a Mogami acquisition than India, Indonesia or Taiwan. Indonesia was offered this ages ago, they had a hard look, and went elsewhere. For reasons. Indonesia just acquired a carrier for a good price. Their closer relationship with the Italians are more than just one platform acquisition. Their needs are also different.
These decisions aren't just made on platform merits too. Its also on geopolitical alignment and situation. Taiwan and Japan aren't perhaps perfectly geopolitically aligned. I think India is a long shot, but India may be interested in Japanese sub-systems, India is pretty strongly now following its own indigenous designs whenever possible. If NZ is waiting for someone else to buy then it will wait too long. Australia has committed to not just buy, but build, that should address any confidence issues any nation has in the design. Realistically, NZ isn't going to use India/Taiwan/Indonesia for a build or upgrades.
Secretly I think Australia would like to keep the Mogami acquisitions to really close allies. NZ is a perfect partner. I wouldn't be surprised if Australia (Gov, RAN, industry) is as hopeful as Japan is of NZ selection. TBH NZ is also a perfect sales partner for Japan. NZ is unlikely to cause international friction due to use of weapons, and NZ is so far away, even China isn't going to be very offended (compared to say a Taiwan sale).
Potential benefits for Australia would also be significant. Training pipelines would be common. Exchanges and mix deployments, would not just be easy, but incentivise crews, because skills are instantly transferable. More career options in that space. People join just because of those opportunities. SOP/operational efficiencies would increase because two navies benching each other with a third. A 3rd nation means another political player to add mass against political decisions to cut upgrades/support for that platform. Interoperability goes way up, munitions acquisitions and stockpiling enhancements. Money goes further. The regional cohesiveness from all three operating the same ship, particularly with other navies. The visual image of AU and NZ being closer together in times of geopolitical chaos. AU/NZ are seen as reliable allies. They are seen as influential allies.
The issues with ANZAC upgrades for NZ should be a clear reminder why something like Mogami would make a lot of sense. NZ would have a choice between two providers for upgrades/services who are operating that exact platform. Both regional players. Both with skin in the game for NZ capability. Both operating large fleets of that exact ship.
For Japan, Australia and NZ are showcase customers. If Mogami just sells to AU/NZ, I think Japan will be very, very happy. That is exactly the kind of countries/industry deals they want. Japan isn't yet going to be chasing middle east/africa sales for 6000t frigates. They want premium customers, who don't come with geopolitical baggage and huge problems. I'm not sure Japan is ready to be dragged through and Indian acquisition process, for example, it could damage them. If they were mildly annoyed and disappointed with Australia's sub selection process, then they would absolutely hate some other processes.