I think if u are doing a institute playthrough u should have li stay in the institute and scara is much better to have join the brotherhood of steel because she is more pro brotherhood and li will hate u because she thinks u tricked her
Li doesn't hate you because you tricked her, she hates you because she hates that every single person and organization she works with eventually screws her in some way. She's a pessimist.
But like App suggested, helping BoS during Institute playthrough probably isn't the best use of time.
Poor Madison Li. I understand what she's been through. It isn't easy being happy when everyone around you is grumpy. Part of the reason I felt really down recently.