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I contacted the CBP through the Trusted Traveler Sentri contact email to ask them. Their answer was short and to the point:
Unfortunately you will not be allowed to enter the U.S. without having your valid New Zealand passport.
I took this period of travel restrictions due to Covid-19 to renew the passports (since I'm currently not allowed to cross the border anyway), and from renewal application to having the new passports in my hand took only 8 days, so we're all ready for when the border re-opens.