I am looking for an answer in terms of mechanics, i.e., spiritual mechanics. Here is my attempt:
For Jesus to take on sin, the Holy Spirit (the deity) disconnected from Jesus' human spirit when he made the first cry. When he made the second cry, his human spirit left his physical body to die. But then, immediately, his human spirit rejoin the deity Holy Spirit. I speculate that Jesus' deity left him between the 1st and the 2nd cry.
Sin cannot be imposed upon the divine nature of Christ. Without carrying sin, Jesus could not have died physically.
Eh...why exactly does Jesus, Who is God, have to obey your made-up rules of when or how He can or cannot die?
There is another answer to this anyway, though I'm not sure how it fits into your "spiritual mechanics" (if it at all), because that's too esoteric to me. Anyway, here is what is proclaimed (in part) in a fraction prayer for the Son prayed in the Coptic Orthodox Church:
"O, Only-Begotten Son, God the Word, Who loved us, and because of His love, He wanted to save us from eternal damnation. But since death was in the way of our salvation, He yearned to go through it, because of His love for us."
In this understanding of what God did and why, Christ's sacrifice upon the cross is the ultimate act of love. Just as He didn't
need to do any of the other things He did (God in Himself is not in need of anything, I hope we'd all agree), He did not
need to die upon the Cross. But He knew that
we needed to be freed from our bondage to sin and death, and so He did so, because of His great love for us.
So there ya go. An explanation (of many, I'm sure) as to how and why Jesus would undergo death that does not in any way undercut His divinity, or separate the natures of which He is composed (God forbid both).