If the costs are close then you’ll have o trade off the comfort of your own bike vs the wear & tear.
Transporting bikes by air can be quite hard on them. Do you have to buy a bike bag?
Transporting bikes around in the back of your car is a headache – they take up space that you might want for other kit.
However much of the riding you might want to do will be nowhere near decent hire shop (i.e. if you have particular requirements on your bike & fit).
Although TBF, I’d just be doing a few days riding form ChCh, Nelson, Alexandra and Rotovegas to keep me happy, lots of bike park and back country to do from there, which could work with hire bikes.