Having a look at redemptions for the three of you one stopping through HKG is 260k points +~£500 in premium and 390k points +~£750 in business. Currently Singapore isn't allowing transit if you've been in the UK in the previous 14 days, and HK not allowing transit if you're on separate tickets. It's these two restrictions that eliminate (as far as I can tell) companion voucher options through these ports as your onward travel would be on a separate ticket.
As you've already got 2 companion vouchers I'm assuming you don't have a way of getting more quickly/easily (through sign up bonuses) and could even *gulp* buy Avios to make up the shortfall (Iberia is selling Avios with a 50% bonus ATM.) The shortfall in Avios could be made up for less cash than your desired total outlay, with the usual caveats associated with this type of purchase.
Later in the year shows daily LHR-HKG-AKL flights and many with Avios availability, however based on their current schedule and the rate of improvement I'm doubtful they'll all be running. Perhaps if booking best to mirror what they're currently operating (or will be in the next month or two).
In any case I feel you'll need an element of good fortune for everything to come off with the move (as does a lot of travel to the antipodes at the moment) but I'm sure with time, flexibility and FT at your disposal you'll get an agreeable solution. Good travels!