@Troisfoisfilles I would do it. It sounds like most things are covered, start little pots for saving towards uniform etc. I would ask the school if they have any additional supports with incidentals, some are very understanding that even those small extras can be challenging. Also many have second hand uniform sales/exchanges.
I was once in a very similar situation regarding one of my dc, and have no regrets. The structure, opportunities, ability to advance and not be bored at school, have prooved wonderful. My other dc aren't in the same sitution, one will never be due to SEN, but even my dc without that those type of schools wouldn't be the right fit. More and more I see that fair isn't equal, each child will have their own needs. My child with SEN I spend money on in different ways - private speech therapy for example.
I would say that there are things to be aware of - some of the kids will come from far more wealthy homes, though not all. In fairness, while we aren't in a deprived area, I actually have met families who are as wealthy in the state sector, and some families were in the same boat as us in the smaller independent school, most of whom were also on bursary and scholarship. Today I see more older cars at our small independent than at the local state primary. Obviously that isn't the case for all, clearly, but not all independent schools are all very wealthy families. In addition, there will be more incidental costs etc. It is just the reality.
Is there any way you can increase your hours if you aren't already full time? Or if you are a teacher, could you tutor? Even just a couple students a week may give you 100-150 month extra. I know it isn't always easy to add in more work. I'm a single parent too and it is a sacrifice. I've learned what times at weekend/after school I can make extra work, mostly work for the family and having those extra funds come in, for me has been a huge help mental health wise, and practially too. Not something I will do forever, but it is really nice to see the emergency fund/buffer build up for incidentals and to know that allows each dc to have their needs met, and opportunities to thrive.