Since you got 6 months old...and given kids to adult ratio is almost 1:1. I will advice to do more relax iteineraries.
day 0: flying from Singapore
day 1: arrive Christchurch - free and easy (take this time to go supermarket shopping and also double check if your SG cloth is enough cause there are too many SG "Winter cloth" look thick but useless as most are cotton cloth.
day 2: drive to Lake Lyndon, Castle Hill, Arthur's Pass, stop at Arthur's Pass National Park Visitor Centre West Coast Road, Arthur's Pass Village, New Zealand for lunch. Then drive to franz josef
day 3: Fox Glaciers Helicopter Flight & Snow Landing
day 4: drive to wanaka via Haast (alot of see along the way)
day 5: Wanaka 4wd Safari and also Puzzle world (good for kids)
day 6: Drive to Arrow town for lunch then to Queenstown
day 7: glenorchy day trip
day 8: TSS earnslaw + walter peak farm
day 9: Queenstown free and easy
day 10: Drive to te anau (evening go for glow warm)
day 11: milford sound
day 12: Drive to Dunedin
day 13: Larnach Castle, Sandyfly Bay, Penguin Place, Royal Albatross Centre etcs
day 14: drive to Moeraki Boulders and then back CHC
day 15: CHC New Brighton Beach, Willowbank Wildlife Reserve, International Antarctic Centre etcs
day 16: fly back to Singapore
Actually since u got 6 mth only... Fox glaciers not a recommend place also I find the Helicopter Flight over price and so-so.
Now already Jul, you need to start booking already...
Also you forgot about Lake Tekapo....