Doubt if you’ll need any other frequent flier programs. These days nearly all airlines have agreements with one another, especially with international carriers.
As soon as you receive your tickets from RPSC call the airline, give them your FF number, and get a seating assignment. Flights fill-up fast. Of course, you can always give them your number when you check-in at the ticket counter.
Mileage wise, I went to the website: http://www.webflyer.com/travel/milemarker/ and figured NYC-LAX-AKL-CHC and it came out to 9433 miles. After that, it’s 2400 more down to McM.
Hope this helps. Safe travels.