as @mom2collegekids says this is exactly it. You cannot reason with this kind of brainwashing and conditioning. Do not try. You cannot actually fault your D for this if she’s not hearing about a potential issue until April, she had no reason to think she would have issues. It is truly unfortunate that you didn’t inject yourself into the process when applications and acceptances were happening and have the money talk then, but there isn’t much you can do about it now, it is what it is.
This is the situation in our household right now and as the stepmom I can very very much empathize with what your wife may be going through.
As stressful and as pointless as you may think it is to fight your ex, you will do far more damage to your marriage, to the ability to finance your other kids educations and your long term financial stability by rolling over now. Remove your wife from it as best you can and protect her from it. It may help you to come vent here, versus to her, so she can feel you have it under control and the ex isn’t winning. This also helps keep you from feeling caught in the middle and pleasing no one. That said, keep her informed, do not make decisions without her input or blessing. You have to be able to fight this together.
As the wife of someone who just sucks it up and pays what he is told the vast majority of the time, his ability to hold firm when he really does want to push back is next to nothing. He does it, much like you, out of fear of alienating his child which I completely understand. Unfortunately all you really do is teach the child they can have whatever they want if they play the game right and show the other children that different rules apply to some. I really do hope you will run a NPC and get a better feel for what you are up against. I strongly suspect that you may not qualify for any aid given what you’ve described, and if you do, it would not be more than 15k which does not sound as if it will be enough.
Don’t risk everyone’s future over this. There are still schools with openings and as everyone has already said, no court will interpret your order to read “any school of D’s choosing at any price without approval from the paying parent”. They just won’t. Modifying it to read in state flagship is a great strategy.