I was the poor kid at the private school.
I got a free place via the 11+ (a very long time ago). All my uniform was secondhand either bought via the school or handed down from older pupils. I didn't go on the ski trips or foreign holidays until towards the end of school when my parents had a bit more money.
Many of my friends (but certainly not all) had enormous houses with swimming pools, tennis courts and paddocks whereas I lived in a modest semi-detached and we didn't even have a car.
I absolutely loved it and didn't care in the slightest that other people had more money and vice versa.Thinking back, I wasn't envious at all...I just felt lucky that I had friends with swimming pools and huge gardens.
However, my sister failed her 11+, went to the local secondary modern which really was a sink school and did very badly. She had undiagnosed dyslexia and struggled until she found her niche and is now a successful writer (ironically). I know that she still has a certain level of resentment that I was the "Golden Child" in our wider family (particularly in my Grandfather's eyes) even if there was not much of a financial difference in the commitment to each of us.
I was lucky that I was in the best place for me but my sisther would not have done well at my school either, she was much more of a rebel than me, much less willing to knuckle under and accept the rules. She would have hated my school just as much as she hated her own and would probably have been expelled fairly quickly.
If I were in your position, OP, I would send your eldest to the private school but make very sure that your younger children do not later feel that they have been judged as deficient in comparison by anyone. When the time comes to decide whether the younger two can go to the same school, make the decision on what is best for them, not what seems "fair".