The Jewish people are said to be Gods chosen and Jesus claimed to be sent only for the lost sheep of the house of Israel at that point, which is probably why He resisted healing her daughter (At first). Even so, she would not let up, and pretty much demanded that He heal her daughter, and much like the unrighteous Judge did when the woman demanded justice, Jesus gave in to the woman because He realized it was the right thing to do, knowing she would not give in. Both woman knew what was right, and what was just and their faith in justice and what was right compelled both Jesus and the unrighteous judge to act accordingly.
Jesus (Although Gods son) was not infallible as far as I can tell. He (Like the rest of us) had to learn and grow in wisdom and in Gods grace. I think the woman lead him to realize what is just and what is not just. This could have been a turning point in His ministry. Perhaps it was this incident that He realized that He was sent for all people and not just the lost sheep of the house of Israel, but for all who were lost.
"God so loved the world ...."
What do "you" think, poison?
First of all, "I" think you need to drop the quotation marks around the word you, implying that my thoughts are not my own.
Something doesn't sit well with the first sentence in your post.
"The Jewish people are said to be Gods chosen and Jesus claimed to be sent only for the lost sheep of the house of Israel at that point, which is probably why He resisted healing her daughter (At first)."
Jews being chosen means we have the responsibility of living according to the Torah, setting an example for the world... being a light to the nations. There is no excuse for ignoring the pleas of a woman in desperate need just because she is not Jewish.
As for "she would not let up, and pretty much demanded"...
And, behold, a woman of Canaan came out of the same coasts, and cried unto him, saying, Have mercy on me, O Lord, [thou] Son of David; my daughter is grievously vexed with a devil.
But he answered her not a word. And his disciples came and besought him, saying, Send her away; for she crieth after us.
But he answered and said, I am not sent but unto the lost sheep of the house of Israel.
Then came she and worshipped him, saying, Lord, help me.
She didn't demand anything. She beseeched him.
She was already humble, and he made sure to humiliate her before granting her request.
But he answered and said, It is not meet to take the children's bread, and to cast [it] to dogs.
And she said, Truth, Lord: yet the dogs eat of the crumbs which fall from their masters' table.
Then Jesus answered and said unto her, O woman, great [is] thy faith: be it unto thee even as thou wilt. And her daughter was made whole from that very hour.
The comparison being made with his metaphor is that the Jews are the children, the non-Jews are the dogs, and the bread is his help/compassion. This is a disgusting incident of racism very unbecoming of anybody, much less someone who is supposedly some sort of great moral teacher.
Yet the dogs eat of the crumbs... several things are going here. She is affirming his subjugation of her by submitting to the dog comparison... and she is also submitting to the notion that her daughter being grievously vexed by devils is a trivial matter not really worth his consideration... like crumbs on a table.
Then Jesus answered and said unto her, O woman, great [is] thy faith: be it unto thee even as thou wilt. And her daughter was made whole from that very hour.
Like a bully who has a kid in a headlock, telling the kid to "say uncle"... and when the kid says "uncle", the bully relents.
Jesus is willing to help her when she declares she knows that her place in the social order is with the dogs who hope to receive the scraps.
I have a few ideas about what's really going on here.
It's possible that Jesus was really was this malevolent, and people identifying him as a figure of love is a tragic mistake.
It's possible that this scenario never happened, but the gospel writers wrote it purposely to convince the reader that this was typical behavior of the Jews of the day, which is why people should embrace Christianity as an open armed, compassionate, loving alternative to a cold, heartless, racist Judaism.
It's possible that it is a combination of the two. Meaning that Jesus wasn't necessarily this cruel, but put on this show in order to attract potential converts by demonizing Judaism, while convincing his Jewish disciples to depart from their ways in favor of a blind devotion to him and his new direction.