Chiming in as Apprentice of the Obvious. To my eye, Kyouko reads as ashamed, above all else. She doesn't think she deserves nice things. She will fight hard, in her current frame of mind, to avoid recovering her friendship with Mami, because that would be a nice thing. Her lashing out is driven from that motivation. That's what we need to break through,
by whichever means, in order to make progress toward anything.
It's an attitude that she's held for a while. Checking up on Yuma can definitely be done without provoking her.
Yuma is innocent and deserves nice things and better people than Kyouko, after all.
It is a decidedly self-flagellating mindset. With a contradiction that I don't completely understand of "I'm a bad person (sinner?) who doesn't deserve nice things" and "I'll look out for myself and take things, as long as they're not too nice". I worry a little about how much of the context is catholic, because I don't understand any religious doctrine very well, much less catholicism in particular. "there's a
place for people like me" seems like an important mantra.
Anyway, the lever I'm
tempted to use to prise open that shell of self-flagellation is Yuma. Kyouko is
raising Yuma, and this doctrine of isolation is bad for her. Give Kyouko a
rationalization to stop this. The best rationalizations are not lies or half-truths, but the pieces of the bigger picture that Kyouko might be prepared to accept right now.
That's not to say that Kaizuki's approach doesn't sound reasonable. In particular, something that provokes Kyouko to try to explain but cast doubt on her reasons for thinking herself such a horrible sinner, seems like a right approach.