Hi! I'm back.
First, I just wanted to clear something up about the Flye Baby. The Flye Baby is
not FAA-approved. It is not in any way a safety device and because of this you can't use it during take off, landing or turbulence. It isn't
prohibited during the cruise portion of the flight, but neither is putting your baby on the floor or walking around the cabin with them or pretty much anything else you want to do with them. It hasn't been tested by the FAA in any way. Baby carriers of any stripe are actually prohibited during take off, landing and turbulence because they have been shown to be -more dangerous- than simply holding your baby on your lap. Just wanted you to know since the wording of the Flye Baby website is (I think intentionally) misleading.
Next, I wanted to let you know that Air New Zealand -only- allows rear facing car seats if they do not prevent the passenger in front from reclining. There is no way you could use the Boulevard rear facing and still allow the passenger in front to recline (which puts you kind of between a rock and a hard place since your daughter doesn't weigh enough to use it forward facing). So if you were hoping to use the seat onboard something like the Coccoro would be your best bet for the 10 month old because at least as far as people have reported here it allows some recline for the passenger in front of it. Having traveled with a 10 month old myself I can tell you that rear facing is so much nicer because you can interact with them and they won't lose their toys, etc. Not to mention she doesn't meet the requirements for a forward facing seat.
Like Ketchupqueen said, if you are up for buying a new car seat for your 10 month old for the trip, I think the Coccoro would be a great choice. It's a great lightweight option and they are even going to make a snap n go stroller for it soon! (There are some cheaper options, like the Scenera, but I don't know how much recline they will allow the passenger in front and it lacks a lot of nice convenience features for international travel.) I would pick either over taking the Boulevard, especially because then if it does get damaged you are not out as much $$$. And some people -have- had success about putting the Coccoro in the overhead bin although there is no way to know for sure before you fly that they will let you do that. Definitely go for gate checking for it though if you can't use it/store it on the plane. This will reduce the risk of it getting damaged. Also, if you check it all the way through as checked luggage you won't be able to use it on the long leg to New Zealand even if a seat opens up on that flight.
Although speaking of seats opening up, if you would need to rebook for your whole family in order to get your daughter a seat I'm assuming it's because the flight(s) you are currently on is/are full. So your chances of their being a spare seat available on your flights doesn't seem very high, especially with the time of year that you are flying.
Given that I might look into this option (if it's available and depending on what other people who have more experience with this seat think): You could buy a Safeguard Go for your older daughter and then put your 3yo in your baby's Boulevard on the plane and carry the Safeguard Go on as carry-on luggage. (The problem is that the Safeguard Go is not being made at the moment and I'm not sure when Dorel is going to start selling the new version, before your trip or after.) But this solution would mean that you wouldn't have to check any seats for your kids and would know that they would arrive safe and useable in NZ. The only thing is you would really want to check on whether the cars you will be using have LATCH/isofix because the Safeguard GO is apparently a pain to install with a seatbelt.
If your older daughter still fits in the Coccoro forward facing and you travel a lot and/or you just feel the convenience would be worth it you could even get the Coccoro, the Safeguard Go, and a CARES harness and then if on your shorter domestic flights there were any extra seats available you could put your oldest in the CARES harness and the baby in the Coccoro, then for the longer flight put your oldest in the Coccoro and have baby on your lap. This would be the ultimate in convenience, flexibilty and portability, but as so often happens also cost. But I just thought I'd throw it out there as a possibility.
A Scenera of Avenue (which ever one your oldest still fits in) would also work for less money than the Coccoro since in this scenario you'd already be committing to not using a seat for your youngest on the Air New Zealand flights.
I would try for one of the last two options if possible to avoid checking any car seats, especially because it seems unlikely at this point that if you bring two you won't have to check one. If you do decide to bring two I would strongly suggest getting the Coccoro because I think it is probably the only seat that will fits your DD that Air New Zealand is likely to let you rear face on the plane if you do get to use it and you have the hope of being able to store it onboard rather than having to check it in the hold. If you do hope to store it onboard try to come up with a bag it fits in that you can pop it in at the gate though.
I hope that helps! And I'm glad you'll consider buying your daughter a seat of her own in the future. I'm a huge advocate for that for the safety and sanity of your baby, your family and all of the passengers on your plane. But hopefully these suggestions will give you some ideas for your options now.