4th Circuit Case probably will be overturned if S.C. takes cert. There is a case on point (somewhat). Rendell-Baker v. Kohn, 457 U.S. 830 (1982). There, the Court ruled a private school wasn't bound by the 14th amendment because there's no state action. Almost all of the private school's operating budget derived from public funds. However, that case is distinguishable a few ways. Most importantly, the case here was about employment practices, not policies concerning student conduct or admissions.
I admit I've only given the current 4th Circuit case a cursory look. I do think charter schools are quite different under the law compared to straight-up private schools. Unless I'm missing something on this issue, if charter schools are allowed to discriminate on this issue, they would be outside the realm of other constitutional decisions such as Brown v. Board of Education. Either the 14th Amendment applies or it doesn't. There's no in-between when it comes to constitutional rights.
I have a question about this. I thought that whether or not a given practice is unconstitutional sex discrimination is constitutional is subject to an intermediate standard compared to a stricter standard for racial discrimination? If so, there would be room to say that separate male/female dress codes are constitutional, while something like separate dress codes by race obviously would not be?
Also, I could see someone from an originalist standpoint taking a absolutist position that the various clauses of the 14th Amendment were enacted to prohibit any sort of racial discrimination at any level of government, but weren't meant to apply to sex discrimination at all. However, that probably only has 2-3 votes even on this SCOTUS.
