Thank you for writing. This is a very complicated situation. I have more questions than answers. Why does your husband need to see his sister once a week? What is going on that he is prioritizing time with his sister on weekends when it is the only time the two of you have together? Why won’t your husband visit his sister’s home? Do the two of you come from such different backgrounds that you have entirely different expectations of marriage?
I don’t think it is expecting too much to want some answers. As I think you know, your main problem is with your relationship with your husband, not with his sister. You and your husband have a very basic and painful (to you) difference of opinion about where his loyalties should be and how much his family should be involved in your marriage.
In western culture, his sister would be seen as violating marital boundaries and your husband would be seen as having trouble shifting his primary alliance from his family of origin to his marriage. I don’t know anything about the expectation of your home country so I can’t comment on that. But the difference between the two may be at the center of your troubles.
What I can comment on is that not coming to agreement about appropriate boundaries could seriously erode your marriage. Your feelings of depression and discouragement are a sign that things are not going at all well.
This cannot be seen as a matter of who wins and who loses an argument. Right now, your marriage is what is losing. If the two of you are to save it and deepen it, it is essential that you work together to decide on how to manage his relationship with his sister. You are offering a compromise of once a month — which he is rejecting. Is he willing to budge at all? If not, it’s important for the two of you to talk about why. Understanding the problem is half the journey to coming to a solution.
If the two of you can’t talk about this reasonably and without getting defensive and angry, I do suggest you see a couples counselor to help you work it out. A couples counselor can provide a safe place for you both to express your feelings and to learn ways to navigate conflicts like this one.
I wish you well.