How about the team rebrands itself to the <INSERT YOUR TOWN HERE> Phoenix, it will take a bit of work but it goes like this.
If you watch from Auckland they play in blue and black..
If you watch from Christchurch they play in red and black.
If you watch from Wellington they play in yellow and black.
And all the branding and posters are tailored to the city that they are advertised in.
Now here's the tricky thing, they actually play in Yellow and Black all the time (shh, don't tell the FFA) but they engage the advanced compositing knowledge of Weta Studio to recolour that Yellow to the colour of whatever city you are watching the Phoenix from. Obviously Sky needs to be involved, but everyone makes a little mess in their pants at the mention of Weta Studio so why wouldn't they be.
We may also have to composite in Eden Park, AMI Park, etc.
I'm assuming that Sky knows where people are watching from and therefore can make these changes, it might require a custom nix firmware on all mysky boxes.
For streaming we can make a good guess at where someone is watching from based on their IP address so can also modify the stream colours in real time.
If people access the Phoenix website it will use the above geolocation to show them their colours and cities name.
If you access it from Australia it will default to the New Zealand Phoenix.
No one will be any wiser.