The scriptures present no requirement by God for an (animal) sacrifice to suffer as part of the sacrifice. However, Jesus whole life was a life of sacrifice, as ours is meant to be (Rom 12:1), and this life required suffering to bring about a specific result - OBEDIENCE!
it was fitting that God, for whom and through whom everything exists, should make the pioneer of their salvation perfect through what he suffered. Heb 2:10
Jesus suffered until his last breath as he fought the desire to do things according to his will and not comply, submit, honour the Father's will for him.
His calling was not meant to be an easy one, but to test Jesus to the the limits of a man who had to, without faltering, put his total trust in his God. This testing naturally requires suffering which calls on self-preservation to avoid pain, humiliation and finally, a gruesome and horrific death.
For we do not have a high priest who is unable to sympathize with our weaknesses, but we have one who was tempted in every way that we are, yet was without sin. Heb 4:15
That Jesus stated he had his own will which clearly differed from his God, speaks to the struggle to remain sinless against extreme temptation and duress.
For I have come down from heaven, not to do my own will, but to do the will of Him who sent me. John 6:38
Having maintained his sinlessness and not succumbed to temptation and resulting sin, Jesus was perfected by his obedience - the suffering was to bring about this obedience.
And having been found in appearance as a man he humbled himself, having become obedient unto death, even the death of the cross. Phil 2:8
During the days of Jesus’ earthly life, he offered up prayers and petitions with loud cries and tears to the One who could save him from death, and he was heard because of his reverence. 8Although he was a Son, he learned obedience from what he suffered. 9And having been made perfect, he became the source of eternal salvation to all who obey him Heb 5:7-
Were the Sufferings of Jesus for the aim of Perfection ?
Yes. The other idea is not presented by the text of Hebrews.
1 Peter 3:18... "died once for our sins, the just for the unjust"
Yes, Jesus died for our sins, but the a requirement for suffering to achieve 'Atonement' (as the OP suggests) was not part of the required death - it was part of what was necessary to bring him to death. The death of itself was sufficient to cover all sin - we are told this in numerous verses - he didn't need to suffer to make this death somehow better!
He needed to suffer, which included his death, as a proof of his obedient and trusting faith in God.
His death solved the whole problem of sin - being raised had little to do with it. Heb 2:14, Rom 5:10, Eph 2:16, Hen 10:20 We live because he lives, but the problem of sin and death and alienation from God was already fixed when he died. That's why he cried out, "it is finished!"
This is borne out by his suffering finishing at death - he does not suffer once risen, exalted, and sat next to God and was glorified. His death, once for all time, applies to all men. The atonement continues until all physical life is changed or destroyed in the second death, Rev 20.
