I would consult with a lawyer specializing in these matters to see what your options are regarding compelling payment from the son’s father.
While I may be wrong, It is my understanding that IDEALLY once both parents agreed on sending a child to private school in divorce cases, both are legally on the hook to pay for it barring some exceptions such as extreme financial hardship…and that doesn’t seem to be the case if the father’s living in a $3 million apartment.
Even with this, litigating cases can take a long time even if you have a strong case so you may also need to examine alternative options including some proposed here by other cc contributors.
Considering he’s a senior who has already graduated and the family doesn’t live in NYC, a nonstarter at this point as application for entry are only accepted in 8th grade for entry into 9th grade or 9th grade for entry into 10th grade.
Not to mention as good as those publics are and despite my being an alum of one of them, those schools aren’t a good fit for every student who qualifies for admission…and that’s assuming they scored high enough on the exam to qualify in the first place.