How do I update payment details for an unrecognized location (Cook Islands) after moving from New Zealand?

Since the Cook Islands heavily shares infrastructure and banking connections with New Zealand, the most seamless option is to keep your Apple Account region set to New Zealand. 

• Payment Details: To update your payment details while keeping New Zealand as your region, you must use a card that Apple can process under the New Zealand App Store. Many banks operating in the Cook Islands (such as BSP or BCI) issue cards that may not register correctly on foreign digital storefronts. If you still have a New Zealand-based bank account or credit card, use that to change your Apple Account payment details. 
• Billing Address: Use your previous New Zealand address or a valid New Zealand address template.