Didn't work - here's the fixed version for anyone interested:

let dateObject = new Date(shippingDate);
let netsuiteFormattedDate = format.format({
    value: dateObject,
    type: format.Type.DATE
});
fulfillmentRecord.setValue({
    fieldId: 'trandate',
    value: new Date(netsuiteFormattedDate)
});

It looks like format.format is returning a string, the string needs casting back into a Date object before passing into the field in NS. I also took the timezone value away and allowed NS to handle that implicitly.