I have a PK who has developed a severe dislike for Sir Kay. Is there any passion he could take besides a Hate(Kay), which may be a bit strong?

Passions tend to be strong. How does this dislike manifest itself? It might be possible to use a directed trait to model it, like:
Disgusted by Sir Kay -X (to Traits in situations where helping, believing or praising Sir Kay is concerned)

So for instance, a normally Modest 16 PK would find himself rolling vs. 11 to keep his mouth shut under Sir Kay's belittling remarks, because he dislikes Sir Kay so much. Or rolling vs. Honest of 5 instead of 10 when asked if he witnessed Sir Kay defeating a giant single-handedly, which he did, but now he might say something like: "I didn't see the whole fight. It is possible that the giant was already wounded from a previous encounter."

I was thinking directed trait Suspicious (Sir Kay) and like you, maybe making it a 5 point modifier