You misunderstand the passage.
Jesus first asserts the primacy of His ministry to the Apostles.
Do you understand what that implies? The Jews were the Choosen People, choosen to receive the the Good News and the first People to receive the invitation to the feast as mentioned in many parables.
She begs for help once again:
Jesus is the Bread of Life, and the children are the Jews, and so the childrens bread is His Word, His ministry; which, as stated above was intended to be preached to the Children of Israel first. You should also realize when taking the full context of His ministry, it was never intended to have the Good News restricted forever to the Jews only, but only while Chrsit was with us.
She understood what He said, even if you do not:
The little dogs are clearly understood by her and the Apostles as any non-Jew, but what she pleaded for and was granted was the same grace and compassion shown by the master (God) of the house (Earth) to the legitimate children at the table (Israel). The litle dogs comment only suggests that all people are within the masters house, not excluded or outside of it.
This is not a case of racism, it is a event to describe the established historical relationship between God and the Jews, and the rest of the world. Therefor it is true when Christ said “Salvation comes from the Jews” for He is salvation and He was a Jew.
Christ knew that after His death, the Word was to be spread to the entire world; not limited only to the Jews, and that it would be a difficult and dangerous task.
If you read the Gospels and see what the direction and conditions given by Jesus of the
first commision to the Apostles, and then compare that to the
second commision, the OP should make more sense.