If you can't afford it don't send your kids there as you risk having to pull them out. Private schools will not bail you out if you run out of money. The bursary is a bit like a scholarship so your kid has to be amazing in a certain subject. And as someone said before, they can help you till the end of the year if something sudden has happened, but running out of savings is not a criteria for getting funding.
I asked about this at the private school my kid previously attended and they said that to apply for financial assistance you have to, quite rightly, disclose all your finances. They come to your house to see what your house looks like and if you can remortgage it. Also, they ask about your parents finances and if they can chip in or if you can take out loans. Also, in this case the bursary didn't pay all the fees even if you did get it, it just discounted them and not by a huge amount either.
If your husband really wants to send the kids to a private school, then consider options for secondary and then see a wealth manager who can advise you how to best invest your savings in order to save enough to pay for them. Patterson Wealth Management do this. I have not used them, but there was a piece on them in the last schools guide, if you are interested: https://nappyvalleynet.com/schools/securing-the-future/
Also, do call the private schools you are interested and ask them what their policy is, but it's not as easy as your husband thinks to qualify for bursaries.