Send a note to NZ requesting a refund to your original form of payment in accordance with both DOT rules and the April 3 Warning Letter. DOT has previously required carriers to initiate their refund within 7 days. Thus, wait 7 days and then file a chargeback with your credit card issuer (bank). If you are denied a refund earlier, then file the chargeback immediately. You should also file a DOT complaint.

The chargeback should include your original e-ticket receipt showing the routing and fare paid, a copy of the cancellation email, and your request for a refund. The same for DOT. This works very well for US banks. Elsewhere, it will be a tossup as to how non-US banks deal with US obligations.

Bear in mind that DOT does not intercede on behalf of individual consumers. Rather, it takes enforcement actions -- sometimes -- against air carriers and will also forward you complaint to NZ with a request for an explanation.