I think he should definitely invite her over when his wife is there, or invite her on a date with both his wife and her, and be honest about it. I feel if she refuses, she's refusing the relationship, she wants to be mono with him and she's a cowgirl.
It's important that he respects his wife's feelings about it. That woman just wants to pretend it's an affair, I feel. It's easier for her to dislike the man's wife and say she's not good enough for him if she never meets her. So it's very important that she does meet her and develop a relationship with her (not romantic or sexual, I mean a metamour relationship). She needs to understand that when you enter a poly relationship, you enter a relationship with all the existing members (unless they want it another way, of course, but in this case the wife definitely sounds like she wants to know and meet that woman).