Whether the Trinity itself was "broken" for a short period of time is, ultimately, unknowable.
I think you are the first to suggest that the Trinity could be separated. While I would understand if one argued for a separation between God and Christ on a physical basis, that separation was possible on an eternal level, from a spiritual perspective, is not something I think many would agree with. I could be wrong about that though.
The Father was with Christ in that hour:
John 16:31-32
King James Version (KJV)
31 Jesus answered them, Do ye now believe?
32 Behold, the hour cometh, yea, is now come, that ye shall be scattered, every man to his own, and shall leave me alone: and yet I am not alone, because the Father is with me.
I believe it was, but on logical ground only (as opposed to textual grounds).
Belief apart from "textual grounds" is not logical.
So, if there is disagreement, it isn't a big issue.
I think its an important issue. That Christ was forsaken is based on His quotation of David in Psalm 22, and the fact is...David was not forsaken:
Psalm 22:23-24
King James Version (KJV)
23 Ye that fear the Lord, praise him; all ye the seed of Jacob, glorify him; and fear him, all ye the seed of Israel.
24 For he hath not despised nor abhorred the affliction of the afflicted; neither hath he hid his face from him; but when he cried unto him, he heard.
Christ also quotes this:
Psalm 31
King James Version (KJV)
5 Into thine hand I commit my spirit: thou hast redeemed me, O Lord God of truth.
And goes on to say...
Psalm 31:22
King James Version (KJV)
22 For I said in my haste, I am cut off from before thine eyes: nevertheless thou heardest the voice of my supplications when I cried unto thee.
However, it is nearly undeniable that the "cup" that Jesus spoke of was the cup of God's wrath.
Christ does not forgive the Father for what He does, but...
Luke 23:34
King James Version (KJV)
34 Then said Jesus, Father, forgive them; for they know not what they do. And they parted his raiment, and cast lots.
Is Jesus asking for another way? Yes.
Again, a point debated in the thread. I take the position it makes little sense for God to manifest in flesh for the very purpose of the Cross and then waver at the end.
I see the Lord as asking that the cup, which entails His Incarnation, His suffering (which itself lends itself to more than being brutalized, but having to partake of human flesh itself), and His atoning for those that made Him weep...pass.
I think it is just that simple: "Father, let this be the time that we get this done."
If we apply "My God my God why hast Thou forsaken Me," then we should equally apply...
Psalm 22:23-24
King James Version (KJV)
23 Ye that fear the Lord, praise him; all ye the seed of Jacob, glorify him; and fear him, all ye the seed of Israel.
24 For he hath not despised nor abhorred the affliction of the afflicted; neither hath he hid his face from him; but when he cried unto him, he heard.
As well as...
Hebrews 5:7
King James Version (KJV)
7 Who in the days of his flesh, when he had offered up prayers and supplications with strong crying and tears unto him that was able to save him from death,
and was heard in that he feared;
Was Jesus forsaken? Yes, absolutely.
Some see it that way. I see that as an impossibility due to the fact God cannot be separated from Himself.
Does that being forsaken necessitate a fracture in the Trinity, perhaps not, but it could.
Just not a possibility.
God is always One, eternally.
Sorry for taking so long to respond, it has been a little busy.
God bless.