Can dad prevent daughter from attending boarding school?

My daughter (D) was 6 years when we divorced in 2008. We have joint legal/physical custody. I’m the custodial parent 5 days/24 hours/ & 7/24, if urgent. At onset of divorce, my D’s well-being was paramount. I provided a daily, steady presence, stable home-life, emotional/physical support because dad works out of town 5 days/week and then takes D from Sat 4 PM through Mon 7 PM, 9 months a year, clockwork. December through February he increases his time with D by 24-48 hours. Been difficult to juggle FT custodial role & pursue career (Nanny) with average wage of $15/hour. Child support is $220/week; my H-E-Loan covers what income & child support does not. My H-E-Loan debt increases at a rate of 12K/year, & there’s not much left to borrow. My D is bright, well-adjusted, mature, independent, in PS 8th grade. D spent 3 summers at music camp living 24/7 w/peers, which piqued D’s interest in Boarding School (BS). In Fall 2013, she begged me to help her with BS application process, aware that financial aid (FA) would be her only “ticket”. When she tried to talk to Dad about her interest in BS he shut her down. She wants to be a neuro-surgeon & study science in an all-girls school where smart girls are celebrated, not scorned. It’s a long-shot but she dares to dream. Dad refuses to submit his financial docs & says “good luck getting accepted without my paperwork.” D plugs away at her SSATs, teacher recommendations, open houses, interviews, essays, emails, handwritten notes. In 98% of cases, a BS will NOT PROCESS an application if either parent refuses to submit financial docs. Admissions results came in today & D has been accepted to 2 top boarding schools (CT & NY) with 75% FA Award! I will do whatever it takes to pay 25% for all 4 years. If she is in BS, I can work full time Nanny in the city and earn up to 80K. Six years of $32/day child support is criminal. Contract deadline is April 10 or scholarship will be forfeited. How do I advocate for her? Can I ask the courts to intervene before April 10 deadline ? Can dad prevent her from attending BS? Please advise.