G’day and welcome! I’m an Aussie (with some pom ) married to a 1/2 Pom 1/2 Frenchman! I spent much of my 20s here and my first 2 were born here. We then went back to Oz when the eldest was 7 and younger 2. After 8 years we decided to come back here for my maternity leave for a year or 2
. 11 years later we are still here!
We are in the Dordogne so also in the Aquitaine. Climate wise quite similar to where I grew up (central west NSW - Cowra / Orange), winters being a bit longer and wetter and can be a bit colder but rarely now. Good plan to come in winter. We did our house hunt in a very cold, wet March so saw everything at its worst (although spring arrived while we were here).
There are not really any truly bilingual schools (perhaps in big cities) and offerings vary. Primary schools tend to be pretty good although English levels of teachers really varies. I would look at what they call private schools here for college (our equivalent of the catholic system, and indeed mostly Catholic, but really not what I know as private). We’ve had our kids in both and while there are some great lycées (15-18), colleges (11-15) tend to be dire. Our youngest just completed his first year of college in one and we are thrilled with it. They also have the option of doing a US high school diploma alongside. For your eldest home schooling may be an option, it is possble, and I think there is a way to do a UK syllabus (don’t quote me on that, something I’ve vaugely heard). Our eldest was 14 when we got back (he’d forgotten all his French as he refused to speak it when we arrived back home) so he did one year and one term at college, then went onto lycée and got a good result. He was somewhere with a European section, they do one subject in English on top if English. Bear in mind it is very, very different, the days are longer (but so are the holidays!), there isn’t as much ‘fun’ in learning here I would say. Also very, very little technology in classrooms, which isn’t necessarily a bad thing. On the other hand I see kids who can barely operate a computer, and that’s in lycée (I teach English).
A few different sides to that, I feel much worse off financially here which impacts in doing stuff with the kids! Also so much is only open in the summer and shuts totally over winter, which we don’t do in Oz.
I’m a midwife in Oz and I was earning around the same each fortnight (11 years ago!) as what I get per calendar month here. Housing costs (out of major cities) are much, much less here and we own outright so that is of course a big chunk of budget we aren’t paying. Food is very similar from what I’ve seen even with all the increases. My son told me of a young French guy who was over there and did a shop in Coles or Woolies and sent his dad a list and got him to buy the exact same food. It came out within about 2€ difference. However it isn’t really as Aussie wages are so much higher so as a % of income its more here.
Fuel is more here. Diesel currently around 2€, so $3. Car insurance cheaper here, and no yearly rego so that side cheaper. Bills to help you make a comparison(we are currently 4-5 people as well), we pay €130 / month in electricity, €60/70 in water (I grow a lot so use quite a bit), 50€ house insurance, about 40€ per car (old deisels), then there’s house tax but that varies hugely by village and house size / type.
You will have more time for the kids maybe, the pace of life is slower I’d say in a lot of ways but your older one won’t finish school until 5pm and they all start at 8am so you’ll find days feel long and busy. Sport is often until 8pm, even when they are little, at least my Remy’s soccer training was!
Social life is very different. Around here it feels like a lot of people either socialise with old friends or family. Often at least one parent seems to work in Saturday so they will only have Saturday night Sunday. There is no popping in for a cuppa or glass of wine like people do in Oz. There doesn’t seem to be much work socialising either. I have some great mates at school but there isn’t much out if work things. I’m quite lonely really and I’m generally a pretty happy go lucky, chatty person. I speak OK French but that is a barrier too as I never feel like I can express myself truly.
I would say if you own and you can manage it keep your house there, or sell and buy something smaller / cheaper. At least until you are sure sure about here being right for you all.
I’m probably being a bit negative Nelly as I am SO over the heat and can’t get motivated to do anything and the holidays are rapidly nearing the end! 

If you have any other questions I’ll do what I can to help!