There's really no need to ask, "Would Jesus have prayed this psalm as a prayer?" We already know He did... from the cross. When I pray and ask would Jesus pray this psalm as a prayer I read every word with that on my mind. What is significant about that with this psalm is something I learned from a friend who is a Messianic Jew. Even though in the text of what Jesus Christ said on the cross was just a part of the first verse as a Jew He implied the whole of the psalm. The previous 24 hours He had served His disciples washing their feet. He enjoyed Passover with them, taught them, warned them, sang with them, prayed with them, for them, and then did something as a human that He hadn't allowed to happen ever. He yielded, gave Himself over for the culture of the Sanhedrin for a mocking trial, then to the Roman culture again allowing Himself to be mocked, beaten, stripped naked to be hung on a cross and be crucified. So with those thoughts in mind read Psalm 22 verse by verse, word by word knowing when He said, "My God, my God, what have you forsaken me?" He meant every word of Psalm 22. He implied that His cry was reminding those present that the mocking by the priests saying, “He trusted the Lord to save Him, let Him trust Him now to deliver Him", that hanging there naked they, could see everyone of His bones, and also obvious right then and there the soldiers were casting lots for His garments.
When David wrote this he may have had little understanding of the implications of what he was writing. But as Jesus did pray this every single word had come true and for the Jewish leaders present, the Jewish people present they knew Jesus meant every word of that psalm. And they could see that every part of that psalm had come true right before their eyes as they stood there. Then like at the tomb of Lazarus when he walked out some believed Jesus, while others ran off to the Jewish leaders to figure out what to do next. Also true like the Syro-Phoencian woman and the Roman centurian whose faith Jesus greatly commended, here at the cross the only one standing and observing who said any positive observation was yet another Roman centurion saying, this was surely a righteous man, this was the son of God. None of the Jews said a word or dared to say a word, even after Jesus prayed Psalm 22 that every Jew present would clearly understand.
To him that excelleth upon Aiieleth Hasshahar. A Psalm of David.
My God, my God, why hast thou forsaken me, and art so far from mine health, and from the words of my roaring? O my God, I cry by day, but thou hearest not, and by night, but have no audience. But thou art holy, and doest inhabit the praises of Israel.
Psalms 22:1 - 3 GNV
And to finish up...
They shall come, and shall declare his righteousness unto a people that shall be born, because he hath done it.
Psalms 22:31 GNV