Hello, I think there are too many harsh replies!
I do think you need to understand more of your SILs point of view, and the obvious, some culture clashes. You say you’re from the same culture, so probs difference of opinions.
Firstly, giving birth and the postpartum period is a very vulnerable time. Understandably, she would only want the people she feels most comfortable with around her. She will be at her most exposed, breastfeeding, emotionally feeling overwhelmed. During this period, it’s natural to only want to live with those whom you can freely be yourself without any sort of pressure, since she will be the one at home most - i.e mother/sister.
HOWEVER, I don’t think it’s right for her to act rude to make your mother feel unwelcome for visiting. I get she would feel uncomfortable if she stayed over, but as long as your mother is being respectful, it’s not fair for her to act rude in response. She could just just grin and bear it, if she hated it that much. After all, it’s her husbands child as much as hers, so his mother, not just a stranger. His sister is also the aunt, as long as everyone is being kind to your SIL, I don’t think it’s fair for her to be rude for them for wanting to see their new niece.
It’s very natural for mothers to feel protective over their new baby, so makes sense, if she was a bit reluctant to pass the baby around. But 3 months later, it’s not fair to not allow you guys to be uninvolved, unless there’s a massive backstory.
I know in some cultures, DILs and sons live with the husbands family, so this is a bitter pill to swallow for you guys. So I understand why your mother felt she was being helpful and meant well by expecting to stay. But it’s perfectly normal and a good thing to want to live in your own space. As long as there’s respect on both sides, I think you should speak to your brother about wanting more access. It’s only fair tbh. After all, you are still family.