Dying is a separation of the spirit from the body. Insofar as Christ’s divine nature was temporarily separated from his body, Christ’s divine nature did indeed die.
But even normal sinful men who die do not destroy their souls by dying.
Nothing of Christ was destroyed by death, except, temporarily, the union of his body and spirit; the first fruits of the resurrection all the dead will experience at the Final Judgment.
What would have been the point of Jesus’ constant prayers to the Father, if the Father and Son were indeed the same person?
Why baptize in the name of the Father, Son, and Holy Spirit?