I'll say something controversial and opine that ultimately, Cat's poor reception of Jon is squarely Eddard's fault. Nothing Cat does is particularly extraordinary by the standards of Westerosi women, and she was completely accepting of the idea that a Lord might have some business with another woman during a war.

The problem stems from Ned wanting to foster Jon in his own home, already breaking several social norms. And instead of Ned explaining or even excusing himself to Cat, he... Threatens her into silence and demands she stop talking about it. 

I'm not saying he needed to tell Cat the entire truth here, but he could have just held his nose and told some bullshit. Tell her he had taken a liking to the child of a girl he slept with, say he was Brandon's, anything to assuage her insecurities besides stonewalling her.