Op it sounds as if you could afford it, and have enough left over for a nice, if materially cautious, lifestyle. Whether your children feel poor or not is largely down to the demographics and wealth culture at the school.
Near me there are a number of fairly average, 20k-a-year-for-senior private schools that seem to attract children from relatively 'normal' backgrounds as well as wealthy ones.
(As a comparison to you, we are on a joint salary of about 85k but with no mortgage. I am quite certain, and this is partly from my own experience as an alumna of one of said schools, that our children would fit in ok at one of these.)
However, there is another, well-known, somewhat notorious public school near us that attracts the offspring of celebrities and oligarchs. Several of my best friends happen to have children there so I hear snippets of what goes on and it does seem as if their kids feel poor there. Which is ludicrous given that their parents are probably on at least your income. It's not an experience I would choose for my children but hey ho different strokes and all that 🤷🏻♀️.
We have opted for state and therefore had enough money to pay for frills. We're in a privileged position of having a choice of several good state schools and colleges on our doorstep. There actually happens to be a fair number of very wealthy families at their school; there is a great diversity of backgrounds and I value their experience of being part of this.
At any private school, there will be moments when you might feel as if you live in a bubble of privileged families where every other child has a pony or goes skiing every winter but you'll have to suck that up and give your head a wobble from time to time and remind yourself that you have chosen to put your children into that environment.
One final word... do not underestimate the cost of teenagers! They are astronomically expensive to keep, even the ones at state schools
and I'd be interested if the people on here who say their children don't mind are actual grabby, materialistic, image-obsessed teenagers yet. so bear that in mind for future budgeting...!