In dilemmas of this kind, always start by thinking about whether they are really inescapable. One escape in this case it to speak with your sister. If she likes New Zealand, she is unlikely to be indifferent to the environmental degradation that is already so much in evidence elsewhere. Plus you can offer to donate the flight cost to a good cause of her choice, in honor of her wedding. In any case, it is much easier for her to understand and accept the decision if she was herself involved in making it or at least in thinking it through.

BTW, I checked your numbers because 4 tonnes seemed like a lot. But you are basically right. A Boeing 747-8 takes a bit over 200 tonnes of fuel (over half its take-off weight), roughly 137 gallons of fuel per passenger. Each gallon produces 20 lbs of carbon dioxide. So that's about 1.3 tonnes per person. But then one tank does not get you there, plus you'll have to fly back as well. So 4 tonnes is a very good estimate. Way too much, indeed.