The Supreme Court's ruling was confined to the meaning of the words woman and man in the Equality Act. It has some knock-on effects, but it does not mean that everyone has to adopt that interpretation of those words in all circumstances. It doesn't even guarantee that is what those words mean in other Acts of Parliament, although it is persuasive.
So no, the SSC ruling does not mean that the word "woman" in the WI constitution can only be interpreted as referring to biological women. That isn't playing legal word games. It is looking at what the Supreme Court has actually decided and avoiding falling into the trap of giving it a much wider interpretation than they intended. If the WI want to interpret "women" as including trans women, the courts will allow them to do so provided the result does not breach the law. However, if they do go down that route, they are clearly not a single sex organisation.
As I've said a few times today, I think there is a decent chance the courts would allow them to be an organisation for women and trans women. I agree it is not guaranteed, but I do not agree with those posters who say the WI is definitely acting illegally. It is possible the courts would find that they are acting illegally, but from experience of how the courts deal with organisations like this, I tend to think that the courts would say they are legal, albeit they are not a single sex organisation.