[nonameiguess]

Really need a Constitutional scholar or attorney to chime in, but as far as I understand, you can base admission to a private club on protected characteristics as well. The cases in which you can't are businesses commonly understood to be public access, like restaurants and barber shops and what not that have street fronts. But Augusta National never had to admit women. They caved to public pressure and Master's sponsors withdrawing money, not to the law.

This is, of course, why all boy's schools and all girl's schools can still exist, too. If HBCUs wanted to formally ban white people, I'm sure they'd face some backlash, but I think it would be legal to do that. All-male priesthoods are still normal and common. The Church of Jesus Christ, Latter Day Saints had an all-white priesthood up until 1978 and that was legal, just another case of responding to public pressure.