There seems to be this idea that principles can only ever be an irritant to you - like they get in the way of you wanting to do stuff and it's better not to have them (or at least put them in inverted commas!)
To me it's not like that - if I think something is wrong, then on principle I am not in favour of it, and therefore I wouldn't want to do it. OP you sound as though you do want to do it (do come back and say more though!), so therefore you don't have that principle any more and it needn't worry you. If your principles were against it, you just wouldn't. Like Nestle, range rover sports and the Daily Mail, I want nothing to do with private education because I don't think they should exist. If you do, that is obviously your prerogative.
Doubtless you will find - as this thread suggests - lots of people who will vindicate your decision and reassure you that you're not a bad person for doing this. And those who would not support your decision, or endorse it, also probably don't think you're a bad person. In the main. There will be lots and lots of people on MN and at the private school who will tell you you have to put your own first, and so on and so forth. If you will find that reassuring enough to make you feel you did the right thing, then that's up to you. It is of course your right to use private schools if you wish.
What you cannot have though is the whole-hearted endorsement of people who genuinely are ideologically opposed to private schooling, and whose principles would not lead them to make this decision. Those people also have a right to their opinion, and you can't expect that they too will tell you 'yes it's fine to change your mind and use private schools, hell, I'd do it too'. I suspect sometimes people won't be happy until everyone concedes that they did the right thing going private, and that in their position, we'd all have done the same.
Not all of us would, and that doesn't make us crap parents, or mean we are 'sacrificing our children on an altar of political correctness'. It just means we have different principles from you.
As the saying goes, you can have what you want most in life: you can't have what you want second and third. So you can have your kid in private school - you just can't get everyone to agree it was the right thing to do. But lots will agree, to be fair, so it's just a question of how much that matters to you.