Australia

There's also the massive general exemption of wether or not your actions are discriminatory.

For example, it's perfectly legitimate to reject an actor who's gender doesn't match the character they are being hired to perform. That is not discriminatory. It's only discrimination if there's no valid reason to reject someone based on gender.

The court ruled, in this case, that there was no valid reason to prevent men from viewing picaso/etc's artwork. "We want to make a point" is not good enough.

I’m not a lawyer, but from what I’ve read, I think if they were to do the paperwork to set up the room as a women-only club somehow, they’d be in the clear.

I'm not a lawyer either, but I don't think the law works like that. Nuance matters in legal proceedings, and finding a loophole like that might reduce your damages but it won't prevent you from losing the case. When two pieces of legislation disagree (e.g. no discrimination vs allowing women-only clubs), it's basically up to the judge to decide which one has priority and they're likely to choose the one that's most important. Which would be the anti-discrimination law, not the club law. The world is full of contradicting laws - judges deal with this stuff every day all day and cases where the law is clear generally don't go to court at all.