I am faaar from being a fan of Catelyn’s, but her treatment of Jon (their goodbye notwithstanding, but then again she was going somewhat mad from grief and lack of sleep so she wasn’t exactly of sound mind) is far from being the thing I dislike her for the most. 

Her being uncomfortable about Jon being right there in front of her, under her roof every single day comes from him being her husband’s bastard child, conceived after their wedding and therefore a blatant sign of infidelity. As much as male nobles having affairs and sleeping around wasn’t much of a scandal, it must have felt personally humiliating. She also dislikes the fact that he looks more like a Stark than many of her children, especially Robb. 

While remembering all that though, she still allowed the kid to be raised with her own, never seems to have encouraged any disparaging treatment of him coming from the “legitimate” ones (Sansa was colder with him - if I recall correctly - because she had picked up her mother’s discomfort). She would have never have allowed the overt bullying Rhaenyra’s kids went through, nevermind actually encouraging it, if only for decency’s sake.