The one place "satisfaction" is mentioned in the Scripture with regard to God is Isaiah 53:
We should be careful not to read in what isn't there. It doesn't say why he shall be satisfied, or what particularly in him shall be satisfied. If it said what you say it says, it would say, "He shall see the travail of his soul, and his wrath shall be satisfied." But it doesn't say that. It just says, "and [he] shall be satisfied".
Could it be that He will be satisfied because by means of His death He will accomplish all the Father's will, namely the undoing of the works of the Devil, the plundering of Hades? By this act, the barren one who did not bear nor travail with child will break forth into singing and cry aloud, and more will be the children of the desolate than the children of the married wife!
God saw of His own labor and was satisfied on the sixth day, then rested the seventh.
What a great reversal! Instead of the lust of the enemy being satisfied upon the people of God, rather the people of God are redeemed by His right arm, pouring out His wrath upon the enemy, satisfying Himself out of this labor, and dashing him to pieces, and dividing the spoil Himself!
In what was the LORD pleased? In the pain of His servant? In the pouring out of His wrath unjustly upon Him... His own arm? His own Son?
Such blasphemy!
No, the very coming of the Holy One constitutes the beginning of the pouring out of the wrath of God upon His enemies and the enemies of His people: namely, sin, the devil, and the final enemy -- death itself!
The last enemy is death.
He was not satisfied by seeing the pain and suffering of His Son twisting in agony on the Cross. He was satisfied by seeing that by this labor and travail, ending in death, sin would be destroyed and death, the enemy of His people, would itself be slain by His life!
No atonement theory that has the Father taking some kind of sick pleasure in the pain and suffering and death of His own Son as punishment or payment for sins in itself (because He needs to see pain because He's been offended) -- for He has no pleasure in the death of the wicked, but that he turn from his wickedness and live! -- is worthy of consideration by any lover of God and His Word.