..you should use the clearly marked, clean and almost-empty family change room rather than the women's change room?
I approached him and said quietly: "Do you know that you are in the women's change room?" To which he replied: "Yes, but she insists in coming in here and the alternative would be a big, screaming fight. I'm actually quite uncomfortable." He then made a swift exit.
Well, guess what- so are all the women who are using the changing room in the reasonable expectation that they would be able to get showered and changed in the mostly shared facilities (just a few cubicles) without being joined by a man.
In his position, I would have the big screaming fight.