Marcus' commentary to Mark 7:27, the parallel passage, is helpful. (Vol. 1, page 469 in the Anchor Bible Commentary to Mark)
For a Jew to equate someone with dogs was an insult. He refers to The Interpreter's Dictionary of the Bible, under "Dog": "The biblical writers ... seem unfamiliar with any kind of warm personal relationship between a dog and its master"
Dogs ate corpses. They were unclean. Dogs in Jewish literature are associated with Gentiles.
We can see that the attitude to dogs was negative in Matthew 7:6, where they are as bad as pigs. (Do not give dogs what is holy; and do not throw your pearls before swine, ...)
Marcus points to literature demonstrating that the normal diminutive for "dog" was κυνἰδιον not κυνἀριον and that in Koine, the diminutive is often used without distinction from the regular form. (my own feeling is: why learn a more unusual inflection of a noun or verb if there is a regular inflectional pattern available?)
It was, no doubt, an insult which only accentuates his appreciation of her faith.