All I really remember about this fic is that rin gets upset that kagome is "replacing" her mother. It's set in modern times, AU, but that's really all I can remember about it. The one specific scene being Kagone giving Rin a gift, but she refuses to open it, resulting in sesshomaru sending her to her room. Kagome feels bad and suggests she leaves, but stays when sesshomaru tells her. It's not much, but I would love if someone knew what story I'm talking about!

I don't know if you're still looking for this story, since it's been a while, but this sounds like a scene in MissKatt's "Behind the Curtain", in chapter 23. (rated MA)

dokuga.com/fanfiction/story/8186/1

All of it seems to fit, I'm just not sure about the part where Rin refuses the gift, but maybe this is a recurring theme eheh.