Let me focus of the gospel account for I think this is more crucial to understand.
O my friend please read the entire account. Our Good Lord didn't ignore her at all. But quite the opposite, He blessed her, He healed her daughter. And not just when He got to it, but INSTANTLY! Why? because of the FAITH she showed(v.28). Faith is what accounts for a persons right standing with God.
Now, why did He call her dog. Let's clarify the context. "And he answered, It is not right to take the childrens bread and throw it to the dogs." Who are the children? the jews/ Israel. Why? because Christ clarifies in the gospels that HIS ministry was to be to the Jews while on earth. Why? b/c of prophecy promised. So, if we understand who His ministry on earth was to be towards, Christ then showed EXTREME mercy and grace by giving His love, attention and energy to this canaanite woman.