I'm coming in kind of late, but I figured I might as well give my two cents.
A nature cannot suffer. If we consider natures to be anything like properties, let's consider something analogous. "I ate the red, but the apple I left aside." That doesn't make sense. The apple truly is red, but you can't eat redness. You eat a red thing.
So a nature cannot suffer. Only persons can suffer. Therefore, either Jesus is God or Jesus is not God. If Jesus is not God, I fail to see why any of you are Christians. If Jesus is God, then truly, God died on the cross.
Last but not least: When it is said that Jesus is fully man, this is not to say that Jesus is only a man. What this means is that everything which is true of man (ideally) is true of Jesus. Every man has a body, a human soul, etc. These things are true of Christ.
This isn't a problem, however, insofar as Jesus isn't a human person, but a divine person.
Christ is not man, who just so happens to be God. Jesus is God become man.
If we go with Sozo's position, we run into the heresy that the Father suffered and died on the Cross, and the Holy Spirit sent Himself into the world.