The response about paring down your fb friends list works only if Kate considers herself *just* a fb friend. If she thinks you're more a real life friend, she'd probably think that sounds logical, and then she'd pursue other means of communication. So it may be wasted effort.
I agree with those saying to ignore anything she sends your way. And if she gets assertive in wondering what the problem is, it may be a fair question. And then I'd tell her the truth - you're not interested in pursuing a friendship with her.