Re. school: can you approach the head and ask for (temporary, you hope) financial assistance? Not all private schools can afford to help, but some can and do. £1400 per term sounds like a pretty good deal to me for a private school, and it suggests that the head won't automatically assume that all parents are loaded down with money. It might be worth an ask. The idea of children having to move schools is horrible for anyone (my great fear in life is that we'll have to take the DCs out of school!!)
As a not-very-wealthy private-school parent, I know all sorts of ways to economise if you do really do want to keep your DCs at their school. In order to keep the DCs at their school, we have no holidays, no 'luxuries' (like hairdressing, gyms, trips to anywhere that costs anything - such as the cinema, theatre, concerts, museums which charge entry fees...), no new clothes (I frequent the charity shops), and an ancient crumbly car (I sold mine when I gave up work). Christmas and birthdays are modest; DH and I set a limit of £20 each last year (so we did things like plant spring bulbs for one another). Our one luxury is an annual National Trust card (which gives us lots of 'free' days out!!)
It does make for a rather curtailed existence by some standards (including my own - I always assumed that I'd have the same standard of living as an adult as I had had as a child!!), but I'd rather the DCs went to their lovely school than I had a haircut or a holiday!
The salary your DH would be getting in his new role is around three times my DH's salary (I am a SAHM), so from my perspective it seems huge. But your outgoings are also presumably far higher than ours at the moment. The good news is that it could well be possible not to move your DCs if you make some cut-backs, even if temporarily. It might not seem like much fun, but it has its own benefits!