that healing just to get rid of her.
That's a pretty definitive statement. He could have gotten rid of her in other ways. He just had to send her away, or get his disciples to send her away, or he could have left the area, or he could simply have said no. He didn't even tell her to go away once, if I recall correctly.
Besides, if group (a) believes in you, tries to be loyal to you, loves and follows you... and group b does none of these things, then who are your 'children' and who are the 'dogs'? It is an analogy. Insulting, perhaps... (though we call one another 'dog' or 'b*tch today in affection). But at least humbling, I suppose. Plus, how does anyone know what people back then thought other people's gods would think of those who didn't believe in those gods? Terms, sayings, definitions, analogies... they change over time, which is why I suggest that you look at both the words, and the deed, to understand.
Christ healed her daughter, despite it all, showing to me that it truly is faith that counts. The promises were/are to Israel, but faith grants anyone else those same promises. Was that the plan from the beginning, or did the plan change... for that you would have to ask Him, yourself.