As a teacher in a private school, I think we are going to see more and more of this situation. We have had several parents withdraw their younger children this year to go to state, to be able to afford to enable their older ones in Years 10-12 to stay and finish their exam years. The VAT hike has tipped the balance between fees being just about manageable with a few sacrifices, to unaffordable for a lot of our parents. So the OP is not alone in having to make this kind of difficult choice.
I think what's important is involving your child in the conversation and helping them to understand what the options are and why you would not automatically send them to the same school as their siblings. An 11 year old is more than capable of understanding that 5 years ago when their older sibling was going into Year 7, the fees were, say, £15,000, but now for them, the fees would be £25,000, and that difference of £10,000 means that the choice is not as straightforward.
My brother and I went to private school but our older sister didn't - she was several years older than us, born when my parents were very young, and they waited to have my brother and I until they were a bit more established. By the time we were of secondary school age, my dad had established himself in his career and wanted to send us to private school. My sister had already finished school by that point and had gone to the local state secondary. There has never been any ill-feeling between us because of that - my sister couldn't care less!
Children can appreciate that when you have children going to school over several years, within that time, family circumstances can change and what was possible for an older sibling isn't always possible for a younger, or vice versa. In an ideal world we want to treat all of our children exactly the same, but life doesn't always work like that.