I am an executive at a private club in WA state.
What sort of club? How is the club organized? Corporation? LLC? Something else?
there is a protective order.
A protective order that says what? How is it that you came to know about this order?
Their divorce is NOT our business
Then why did you tell them this: "We have told the couple, that if HE is in the bar first, she cannot enter. And if SHE is in the bar first, that HE may not enter"? You kinda made it your business by doing that.
Now comes her lawyer. The lawyer is advising HER to enter the club, and if HE does not leave to call the cops and have him arrested.
How is it that you purport to know what her lawyer told her?
By the way, does your private club have an attorney? If so, have you consulted with him/her about this?
Where are our rights in this?
As phrased, this question makes no sense. However, that's partly because you provided no relevant information about the protective order.
How would we know what the restraining order says?
That's a good question, but your initial post indicates an awful lot of knowledge about the situation.
Can we refuse to let her in is HE is there? This is not a public club, but both are members.
Your club's rights and obligations vis-a-vis its members are, presumably, spelled out in the membership agreement, club by-laws, and whatever other documents govern club operations.
We can't throw them out for doing things that are legal can we?
Can we cancel their membership?
How could we possibly know?
One final point: the protective order presumably imposes certain obligations and/or restraints on the parties. However, it likely does not purport to impose any obligations or restraints on third parties (such as you and your club). In order for it to do that, a third party would need to be joined in the action in some manner. If that hasn't happened, and if the protective order hasn't been served on a third party, then the protective order is a complete irrelevancy to the third party.