My understanding is that, because Jesus was fully human (never forgetting that he was fully divine), he had, like all humans, to freely choose to love God even while carrying his cross.
Now, and this is entirely me thinking about this and not backed by anyone that I know of who has any authority to say something like this so please correct me instantly if I am wrong, but does it not make sense that for Jesus to experience the full human condition he had to have a “dark night of the soul”? (A spiritual crisis in a journey of faith…)
And is it not all the more beautiful, then, that in spite of this experiential loss of contact with divinity (though he was still in fact fully divine) he perseveres his complete trust in God by saying, “Father, into your hands I commend my spirit?”
How many of us, when carrying our own crosses, are suddenly faced with our own perceived lack of connection to God? Can we still say, during those times, “Father, into your hands I commend my spirit?”
One of the reasons upheld by the Church for the Incarnation is so that Jesus could show us in his teaching and by his example how we are to live as God’s people. This was all to say how I understand that scriptural passage.
As to your question about spiritual death…
You are right that spiritual death is separation from God. However, it is a result of sin, of turning away from God. Jesus, even in his humanity, did not turn away from God. Jesus, in his divinity, could never be separate from God, as he IS God. He could not have died the spiritual death that we choose by rejecting God.