From a trinitarian perspective, how do you biblically reconcile Jesus dying and Jesus being God?

You are asking about the "Trinitarians" view, not about a certain denomination. Because there are many different Trinitarian denominations, various different answers to your question exist.

God can't die, therefore Jesus isn't God.

Jesus dying ... and God ... dying

In my youth I heard about various books written by Christian theologians arguing more or less this way:

"One reason why God became a man is that he wanted to die because he wanted to make the experience of dying."

These authors deny the premise ("God can't die") of the non-Trinitarian's statement (see the second part of my answer).

I don't know which denominations were represented by these theologians, however, it is clear that the statement implies that those denominations at least assume that Jesus is God.

First of all, does the Bible even say that God can't die (I assume it does, or at least it implies it).

You have to be very careful here:

If you assume that there is no life after death, the sentence "some person died" is equivalent to the sentence "the person does not live any more".

However, the Bible assumes that there is a life after death. And if a religion (not necessarily Christianity) assumes that there is a life after death, the sentence "the person died" does not imply that the person does not live any more. The statement that somebody has died does not even imply that the person is dead (even not for a short moment) after having died.

And in reverse, the statement that some person will live forever does not imply that the person does not die.

Unfortunately, I cannot tell you if there is some text in the Bible explicitly saying that God cannot die.

However, the quotes from the Bible given in the other answer only say that God will live forever; assuming that there is a life after death, this does not imply that God cannot die.

You answered your own question a bit: It was Jesus' body that died. When we refer to someone dying we usually mean one's physical body is dying. Atheists then think the human is "entirely" dead (no longer existent). In fact, nobody "dies" like this, just their bodies. The deceased then go to Heaven, hell or purgatory.

Jesus' body was dead for two days. After the death of Jesus on the cross Jesus went into hell, reunited with His body on the third day and rose to Heaven.

In the Apostolic creed:

I believe in God, the Father Almighty, Creator of Heaven and earth; and in Jesus Christ, His only Son Our Lord, Who was conceived by the Holy Spirit, born of the Virgin Mary, suffered under Pontius Pilate, was crucified, died, and was buried. He descended into Hell; the third day He rose again from the dead; He ascended into Heaven, and sitteth at the right hand of God, the Father almighty; from thence He shall come to judge the living and the dead. I believe in the Holy Spirit, the holy Catholic Church, the communion of saints, the forgiveness of sins, the resurrection of the body and life everlasting.

(emphasis mine)

In Scripture, sometimes with "death" also spiritual death is meant, but that's a different story that doesn't play a role in this case.

The Doctrinal Position

Trinitarian position usually includes the doctrine of Jesus's having two natures in one person (Hypostatic Union). Armed with both the doctrine of Trinity and the Hypostatic Union, the standard explanation of what happened when Jesus died on the cross is as follows:

1. The Divine nature of Jesus did not die or cease to exist
2. Neither God the Father nor God the Holy Spirit died on the cross
3. The body of God the Son in his human nature died and was buried
4. Although the human body of God the Son died, the hypostatic union of two natures was never separated, broken, or compromised

Source: Did God Die on the Cross?: The Trinity and the Crucifixion

Answering your questions

Q1: Does the Bible even say that God can't die?

Answer: Yes, see several Bible verses such as

• Isa 40:28: ... The Lord is the everlasting God, the Creator of all the earth. ...
• 1 Tim 1:17: All honor and glory to God forever and ever! He is the eternal King, the unseen one who never dies; he alone is God. Amen.
• Deut 33:27: The eternal God is your refuge, and his everlasting arms are under you. ...

Q2: [paraphrase] How do Trinitarians explain that Jesus truly died as a human being but yet retains His Divine Nature who never die?

Answer: Quote from the article above for the explanation:

As mentioned above, human nature doesn’t cease to exist in death; rather, the body perishes but the soul/spirit lives to God. Jesus’s human nature—like ours—still existed in his death, because the soul/spirit is immortal and thus the human nature still lives in/not in the presence of God. If Jesus’s human nature died/ceased to exist for three days, this would indicate not only a death of his soul, but also a split in his person—only half of Jesus would exist for three days while his body was in the tomb. We need to affirm, then, that the human soul/spirit of Jesus remained alive (thus, his nature did not die), but that he experienced a real human death like all of us: body in the ground, soul/spirit with the Lord. And his resurrected body, like ours one day, was raised imperishable and he now lives as the God-man who will never die again.

Q3: What's the Biblical evidence for the answer of Q2?

Answer:

STEP 1: Trinitarians look at the Bible verses below as the starting point (with straightforward plain sense interpretation).

• Jesus is God: see supporting Bible verses such as
  • John 1:1,14: In the beginning the Word already existed. The Word was with God, and the Word was God. ... So the Word became human and made his home among us. He was full of unfailing love and faithfulness. And we have seen his glory, the glory of the Father’s one and only Son.
  • John 10:30: The Father and I are one.
• Jesus died: some supporting Bible verses:
  • John 19:30-33: When Jesus had tasted it, he said, “It is finished!” Then he bowed his head and gave up his spirit. It was the day of preparation, and the Jewish leaders didn’t want the bodies hanging there the next day, which was the Sabbath (and a very special Sabbath, because it was Passover week). So they asked Pilate to hasten their deaths by ordering that their legs be broken. Then their bodies could be taken down. So the soldiers came and broke the legs of the two men crucified with Jesus. But when they came to Jesus, they saw that he was already dead, so they didn’t break his legs.
  • Mark 15:37: Then Jesus uttered another loud cry and breathed his last.
  • 1 Pet 3:18: ... He suffered physical death, but he was raised to life in the Spirit.

STEP 2: Trinitarians then came up with the doctrinal position above as the straightforward logical way to reconcile them, yielding the explanation in the answer to Q2 above.

Conclusion

The straightforward meaning of the Bible verses and the explanation preclude having to "grasp for straws".