If the membership is private, then it is NOT an asset (like a car) that anyone can go get. The membership is conferred by commitee based upon whatever their criteria are, anhd not open to any member of the public. And that membership was conferred to him. That is why I believe the club would need to agree to transfer the membership to another party, and I believe the court ALONE cannot impose that decision against the clubs wishes (if they feel he is more of an "asset" to them).
I have no idea who brings in members, does more to publically promote the club, and does what all the club likes to see on their behalf. But they MAY have their own ideas on this. I know of some clubs that have large charity golf outings held there specifically because they have a member Rotarian or charity Board of Director who arranges for the club to host that event, as an example.
Also, if he purchased the membership for business purposes, and she does not need it for that, then the membership is obviously more important to him than her. Which is something to consider if any CS or alimony is being sought, as losing networking can impact business relationships.