Why is it necessary to carry off a racial slur before her daughter is healed?
This is not a racial slur. Jesus is not referring to her ethnicity. Rather Jesus is addressing her religion and status as a gentile outside the covenant of Israel.
The woman is being referred to as a non-Jew, a Greek.
Mark 7:24 And from thence he arose, and went into the borders of
Tyre and Sidon, and entered into an house, and would have no man know it: but he could not be hid.
25 For a certain woman, whose young daughter had an unclean spirit, heard of him, and came and fell at his feet:
26
The woman was a Greek, a Syrophenician by nation; and she besought him that he would cast forth the devil out of her daughter.
27 But Jesus said unto her, Let the children first be filled: for it is not meet to take the children's bread, and to cast it unto the dogs.
28 And she answered and said unto him, Yes, Lord: yet the dogs under the table eat of the children's crumbs.
29 And he said unto her, For this saying go thy way; the devil is gone out of thy daughter.
Matthew 15:21 Then Jesus went thence, and departed into the coasts of Tyre and Sidon.
22 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.
23 But he answered her not a word. And his disciples came and besought him, saying, Send her away; for she crieth after us.
24 But he answered and said,
I am not sent but unto the lost sheep of the house of Israel.
25 Then came she
and worshipped him, saying, Lord, help me.
26 But he answered and said, It is not meet to take the children's bread, and to cast it to dogs.
27 And she said, Truth, Lord: yet the dogs eat of the crumbs which fall from their masters' table.
28 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.
What I see here is yet another example of Jesus embarrassing and showing up the chosen people by drawing attention to the faith of gentiles in recognising Jesus as one
sent from God and worthy to be called Lord. There are plenty of people
who by faith came into the nation of Israel (Ruth and Rahab for example).
John 1:11 He came unto his own, and his own received him not.
This was an ongoing problem throughout the gospels and Jesus often had to challenge and rebuke the Pharisees, Sadducees and Scribes responsible
for implementing the Law in Israel at this time for their unbelief.
Matthew 8:5 And when Jesus was entered into Capernaum,
there came unto him a centurion, beseeching him,
Mat 8:6 And saying, Lord, my servant lieth at home sick of the palsy, grievously tormented.
Mat 8:7 And Jesus saith unto him, I will come and heal him.
Mat 8:8 The centurion answered and said, Lord, I am not worthy that thou shouldest come under my roof: but speak the word only, and my servant shall be healed.
Mat 8:9 For I am a man under
authority, having soldiers under me: and I say to this man, Go, and he goeth; and to another, Come, and he cometh; and to my servant,
Do this, and he doeth it.
Mat 8:10
When Jesus heard it, he marvelled, and said to them that followed, Verily I say unto you, I have not found so great faith, no, not in Israel.
Mat 8:11 And I say unto you, That many shall come from the east and west, and shall sit down with Abraham, and Isaac, and Jacob, in the kingdom of heaven.
Mat 8:12 But the children of the kingdom shall be cast out into outer darkness: there shall be weeping and gnashing of teeth.
Mat 8:13 And Jesus said unto the centurion, Go thy way; and as thou
hast believed, so be it done unto thee. And his servant was healed in the selfsame hour.
Jesus came to save people, not to slander or denigrate them.
But he also threw back upon the chosen House of Israel, - those who were chosen and under the Abrahamic Covenant, who had received the Law through Moses and the Prophets,
who should have known better on the righteousness of God and the promise of a Messiah - that they had failed to live up to the legacy given to them.
Romans 11:11 I say then, Have they stumbled that they should fall? God forbid: but rather through their fall salvation is come unto the Gentiles, for to provoke them to jealousy.