I think your DH is right about the private schools. I think I do, anyway. It definitely needs thought.
Your kids are going to be living in a massively more expensive home than many of their friends. I wonder how that will play out. Will it be awkward for them?
If you decide you don't like aspects of the state school and you don't have time to micromanage and improve the experience by liaising with school or becoming a school governor - then will you be able to afford private schools after all if you decide you want to later, and you've purchased this costly house? . The older children get, the more issues can impact their school experience.
I believe in state education, but I do accept it is hard going through a school and being different to the others. Easier if your family is middle class and socialist, as are many families in SE London. Especially if your parents are seriously well off and live in a house that you could buy maybe three small houses or nice flats for in nearby areas, it could be a bit tricky. . Think of the kids' potential experience.
You say you have not been in London long, perhaps you will not be offended by me commenting this way, as I wouldn't like you not to know about the relative pros and cons of schools.
It strikes me that the house you post, at that price, is ostentatious and grand. Do you really need that? Why not save on your mortgage, because who knows what might happen in the job market.
I even think your DH is right about watching prices, as to fall by 5 - 10% would be a massive life- impacting saving for your future.
Look for s cheaper area and save on your mortgage. Aim for 800 or 900k . And research the ethos of different schools and even the emphasis in the teaching and how certain issues are dealt with, say in 'An Inspector Calls' , the GCSE English Lit text