Welcome to the community. The one doctor's recommendation to not travel did seem extreme. Cyst ruptures can be quite painful (although I have not personally experienced a rupture). If your pain becomes severe while you are gone and you don't have effective pain meds, it would seem reasonable to go to a New Zealand ER. Also, 12 weeks seems a bit long to wait for another ultrasound. 6 to 8 weeks seems to be the norm with a complex cyst this size. It may go away on its own but it could grow rather quickly too.
Hope your trip goes well in spite of this cyst.
Also the first doctor said I should go to a gyno and the second doctor said he wouldn't worry about it because they will say they same as the ultrasound report (come back in 12 weeks )