It is very important for your respected wife to educate herself about the definition and rulings related to maḥram/non-maḥram cases.
Unfortunately she is misinformed, and has only half of the facts correct.
In Islam, a man cannot marry two sisters at one time, but this does not mean that the wife's sister becomes maḥram to the husband, nor does it mean that the sister's husband becomes maḥram to the wife's sister.
So, although he cannot marry his sister's wife as long as he is married to his wife, it still does not mean that she becomes maḥram to him.
It is still haram for him to see her without hijab, or look at her with lust, or shake her hand, or be alone with her, and so on.
This is a view that all our jurists have.
As for the case of her staying at their home, if all other shar'i matters are observed, yes she can.
And Allah knows best.