Ah, not failing actually. Your claim is that Jesus two natures implies that His divine nature dies and reason it has to endure is because it is divine. But Jesus experienced death in his humanity, and thus because He is one person this is communicable in value in the person of Jesus.
If only a man died on the cross. A man has only the value of one man however good.
What happened is the reality that two natures are communicated. Since He is truly one person, what he experiences by his humanity is experienced in his person hood. If this communion does not exist, than God the second person does not assume the humanity of Christ. If God the second person of the trinity does not, neither is his death on the cross efficacious.
The reality is that there is a communication of the two natures to the one person. What was called a "hypostasis" by the Council of Calcadon in 451. It appears also in Heb 1:3.
It is because of the God-man value of the person of Christ that the cross is efficacious. If only a man died on the cross. A man has only the value of one man however good. Only because a communicable divinity and humanity existed in the one person of Jesus does the sacrifice have its value.
Divine nature does not die. However, the Divine nature is also said to be one nature of Jesus by the incarnation.
Consequently, this is the what the incarnation means. Jesus who is God assumed his Human nature by his incarnation and became Jesus both God and Man. His humanity is inseparable from his person because he has assumed it. His divinity is inseparable from his humanity because they are both existing in one person who has both natures.
But because the incarnation means that Mary is the Mother of Jesus the incarnated person, she is also the mother of God. Because the incarnated person is also God. This means it is precisely by the incarnation that Mary really is the Mother of God.
If your claim is that by saying that Mary is mother of Jesus equals Mary being mother of God, then y’all are making NO distinction between the incarnation of diety and Diety Himself.
Therefore by your claims and failure to make the distinction, God is both human and Deity, and when Jesus died on the cross, then Deity died.