That's not the basis on which Labour are campaigning. The stated goal is to raise £1.7bn. The assumption is no children will move across, "the rich will pay even more, they always do". Starmer says "I want private schools to thrive". Reeves says "all my policies are fully funded".
Perhaps you're right, and they should campaign on the basis of harming private schools, hang the economics, and the "additional cost will just have to be found". That is, at least, the reality as many of you on here seem to agree.