This is very sad, especially in relation to how you have described your SD in earlier days. As advice said, the child is being set up for massive failure. Very disheartening. 

Giving me flashbacks!  BM sounds exactly like the Girhippo.  Schoolwork, homework, attendance was completely optional because "trauma of divorce."

All 3 skids had an IEP although the only "disability" they had was p!ss poor parenting.  Because the Gir is pratically the mayor of her town, the school gave up while vaguely implying that 100% custodial Gir wasn't enforcing anything.

Private schools are not likely to accept a kid with character and performance issues.  Some, if not most private schools have strengent acceptance criteria including placement testing.  Our son was a marginal student in Middle and High school.  When we decided to pursue a boarding school for him we toured several.  The one we chose did not just take anyone whose parents could cut a check.  They interviewed us, they interviewed him separately, they put him through half a day of academic testing.  

He did incredibly in that environment. Until he didn't. We refused to pay crap tons of money for him to not deliver. So after a year as an honor student, highly awarded cadet, etc... he started his Sr year. His Spermidiot had hacked the school firewall and SS and dipshitdaddy would stay up all night playing WoW. They even had other cadets doint it. So, we withdrew him at winter break, brought him home, and rode him like a broke back mule until he graduated on time and ultimately with honors from our local HS. He knew no one there. We had moved shortly after he started at military school.

He is 33, successfully serving in the military, is 5yrs & 46 days from reaching full retirement eligibility.  He has told us a number of times that he regrets blowing his opportunity to graduate HS from and return to his military school for JC and an early military officer's commission.   Regardless of the quality of the opportunity, the kid has to perform.  

Our dear friends went to the same process with their son at the same military school.  The school declined to extend acceptance to their son.  He had behavioral issues, blew the interview, and bombed the academic testing elements of the application process.

I would make it clear to BM that tuition is on her and that CS is dad's share of any kid related costs.  I hope that your SKid steps up and delivers.  But it is probably wise to lower your expectations, so you won't be disappointed.