When asking "would Jesus pray this psalm" it really isn't necessarily a Yes or No answer. You are asking to learn the humanity of Christ. How much did He depend as a man upon His Father and not His own Deity. It helps explore Christ in a new an fresh way, especially as His example of dependence on His Father is necessary to see, experience, live in Him.
If ye then be risen with Christ, seek those things which are above, where Christ sitteth at the right hand of God.
Colossians 3:1 GNV
For Jesus His Father was above. Today Jesus is also above. When Jesus prayed His high priestly prayer in John 17 that which He prayed to His father was for a very few specifics. First that Father would keep them (us) from the evil in the world, and that the Word He had given them would remain in them (us). He knew what they needed for His people ahead.
In our church service today a couple had been asked to share a testimony of what took place in their lives. When they woke up one morning the husband was barely breathing. They rushed to the hospital emergency department. When the wife had to insist that CT scan had to be skipped, the attending doctor stepped in, agreed, and took charge immediately getting a breathing tube in so he could breath. For 5 days the hospital staff continued doing all possible. The right surgeon was there at the right time capable of performing a needed surgey. He later said it was the most difficult surgery like that he had ever done. In the end the husband was able to leave the hospital, and was indeed there standing before us this morning. When looking back they could see God's working in situations through the hospital staff around them and also using people from the church close to them
As I listened to this testimony I thought back to my wife laying in the hospital three weeks as we had hoped at first she would soon leave, when in fact a second hemorrhage occured when they were going to do surgery so she could be moved. So instead of the helpful surgery she had emergency surgery to do the best to save her life whatever that might end up being. When I learned the extent of the damage, then all our children, my children said it would be her desire to have life support removed. Agreeing with them shortly we walked, I walked out of the hospital without her. My result was very different from the couple from this morning.
This psalm begins by describing a battle in which more than 12,000 of the enemy died. Jesus never experienced that. But just before telling that though it is described as a psalm of David for teaching. It wasn't only for us to learn from, but also something from which Jesus was to learn. It was something He knew because of His deity, but it would be something else to know experientially as fully man. This other couple went through was something Jesus did not go through. Even my experience with my wife was not something while on this earth that Jesus went through. Both of those situations having to do with a spouse. But of those things, significant victory in battle, near death of a loved one, and death of a loved one itself were all things Jesus experienced. We know He prayed before His most significant greatest victory as we see Him in great disress both human and of His divine nature in that prayer time, praying, laboring in prayer three times before submitting to the cultures that would take Him to the cross and He )commending His spirit to His Father, that same Father who would raise Him to life again.
But now thou hast given a banner to them that fear thee, that it may be displayed because of thy truth. Selah.
Psalms 60:4 GNV
Yes, this psalm not only tells of how God gave great victory, but tells how He chooses to proclaim to whom He has given that great victory to as well. His banner is to them that fear Him, and is displayed because of truth. God has a symbol that can be seen because He gives us the victory, that victory is to them that fear Him, again displayed because of truth. This brings to mind another time we read about His banner identifying the believer.
He brought me into the wine cellar, and love was his banner over me.
Song of Solomon 2:4 GNV
Because of the banner of Psalm 60 being given to them that fear Him it is fitting for Solomon's Song being reflective of that same relationship. The very real struggles that bring us to him as drastic and life changing they may be, that reward of that banner of His love over us displays whether we feel like it or not. It is God's victory, His claim, His banner over us, His love, His truth.