As I understand it, there are five Psalm 22 verses that are alluded to in the Gospels. If I missed any, please let me know.
Preliminaries
Jn 19:23–24 - When the soldiers had crucified Jesus, they took his garments and divided them into four parts, one part for each soldier; also his tunic. But the tunic was seamless, woven in one piece from top to bottom, so they said to one another, “Let us not tear it, but cast lots for it to see whose it shall be.” This was to fulfill the Scripture which says, “They divided my garments among them, and for my clothing they cast lots.” So the soldiers did these things.
Mt 27:35 - And when they had crucified him, they divided his garments among them by casting lots.
Mk 15:24 - And they crucified him and divided his garments among them, casting lots for them, to decide what each should take.
Lk 23:34 - And Jesus said, “Father, forgive them, for they know not what they do.” And they cast lots to divide his garments.
2. Psalm 22:7 —“All who see me mock me; they make mouths at me; they wag their heads”
Mt 27:39 - And those who passed by derided him, wagging their heads
Mk 15:29 - And those who passed by derided him, wagging their heads and saying, “Aha! You who would destroy the temple and rebuild it in three days
Mt 27:43 - He trusts in God; let God deliver him now, if he desires him. For he said, ‘I am the Son of God.
Mt 27:46 - And about the ninth hour Jesus cried out with a loud voice, saying, “Eli, Eli, lema sabachthani?” that is, “My God, my God, why have you forsaken me?”
Mk 15:34 - And at the ninth hour Jesus cried with a loud voice, “Eloi, Eloi, lema sabachthani?” which means, “My God, my God, why have you forsaken me
Jn 19:30 - When Jesus had received the sour wine, he said, “It is finished,” and he bowed his head and gave up his spirit.Psalm 22:31— For dogs encompass me; a company of evildoers encircles me; they have pierced my hands and feet
Luke 24:39 - See my hands and my feet, that it is I myself. Touch me, and see. For a spirit does not have flesh and bones as you see that I have.
Note: Some say that that Psalm 22:16 should read, “like a lion, they are at my hands and feet.” However "like a lion," and "they have pierced" are very similar in Hebrew: כּארי verses כּארוּ. And the older Hebrew manuscripts, and manuscripts in other languages that predate most of the Hebrew manuscripts, strongly argue for “pierced” being the correct reading.
Conclusion: As I said, there is no instance where PS 22 was copied word for word in the Gospels. The ideas in PS 22 verses were certainly used by the Gospel authors in their own words.
So is there a better explanation for the use of PS 22 other than some sort of fakery?
Yes. Jesus knew the OT and taught it to the Apostles, who also mostly likely read the Scriptures daily. And most likely Jesus taught this Psalm in regard to his death. So, the Gospel writers had that teaching blazing in their hearts and minds when they realized what happened and was said during the crucifixion, thus fulfilling PS 22. So, they wrote down what they heard and saw. No fakery or forgery needed to explain the allusions to PS 22 in the Gospels.
There are Jewish writings which associate this Psalm with a future event and Messiah.
Commenting on this Psalm, Rabbi Rashi [Shlomo Yitzchaki] says, ‘They (meaning the Jewish people) are destined to go into exile and David recited this prayer for the future.’ So this would mean that the Psalm does have a future application.
Rashi’s commentary on this verse was: “Why have You forsaken me?: They are destined to go into exile, and David recited this prayer for the future.”
The famous Midrash from the eighth-century, Pesikta Rabbati, places some of the words of Psalm 22 on the lips of the suffering Messiah. In fact, the Midrash explicitly states that: “it was because of the ordeal of the son of David, that David wept, saying: “My strength is dried up like a potsherd, and my tongue sticks to my jaws; you lay me in the dust of death.” (Ps. 22:16)
According to this Rabbinic Midrash, King David described the future suffering and death of Messiah Son of David in this Psalm.
The following rabbinic Midrash which was written prior to the Masoretic text:
“During the seven year period preceding the coming of the son of David, Iron beams will be brought low and loaded upon His neck until the Messiah’s body is bent low. Then He will cry and weep, and His voice will rise to the very height of heaven, and He will say to God: Master of the universe, how much can my strength endure? How much can my spirit endure? How much my breath before it ceases? How much can my limbs suffer? Am I not flesh and blood? …During the ordeal of the son of David the Holy One, blessed be He, will say to him: Ephraim, My true Messiah, Long ago, ever since the six days of creation, thou didst take this ordeal upon thyself. At this moment, thy pain is like my pain. At these words, the Messiah will reply: ‘Master of the Universe, now I am reconciled. The servant is content to be like his Master.'” [Midrash Pesikta Rabbati, 36:2]
The Midrash goes on to clarify:
“Ephraim, our true Messiah, even though we are thy forbears, thou art greater than we, Because thou didst suffer for the iniquities of our children, and terrible ordeals befell thee. For Israel thou didst become a laughingstock and a derision among the nations of the earth; And didst sit in darkness, in thick darkness, and thine eyes saw no light and thy skin cleaved to thy bones, and thy body was as dry as a piece of wood; and thine eyes grew dim from fasting, and thy strength was dried up like a potsherd (Psalm 22:16), All these afflictions on account of the iniquities of our children.” [Pesikta Rabbati 37:137]