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Following Tim's response above...
It doesn't appear that the woman would have taken it as an insult.
The first thing to note is the Greek word for "dog" that is used refers to a "little dog" or a household dog. This is different than the scavenging dogs that roamed free and ate "unclean" things. The NKJV version captures this and translates it "little dog."
The fact that the woman extends the metaphor in her response to Jesus (Matt. 15:27) shows that she recognizes the analogy and her place as a gentile within Jesus' current mission.
She is clearly familiar with Jewish customs and the cultural thinking about the Messiah because she refers to Jesus as "Son of David" - which was a Messianic title during that period (Matt. 15:22). As a close neighbor to Israel, she would be familiar with Israel's God and His Messiah.
God loves persistent faith! Those who know the claims about the person of Jesus and do not stop as they seek blessings.