Bryan's argument was provided a variety of different ways with many different tangents, but Bryan's premises can be reduced to this:
1. Penal substitution requires Christ being punished by God.
2. Punishment requires a loss of communion between God and Jesus.
3. A loss of communion between God and Jesus means either that Jesus is two persons (one person who is God and one person who is man), that Jesus is not God, or that there are more gods than one. (Respectively, those positions would be identified as Nestorianism, Arianism, or Polytheism.)
Penal substitution requires Christ being punished by God.
We don't object to Bryan's first premise. Isaiah 53 teaches this. That chapter states:
He is despised and rejected of men; a man of sorrows, and acquainted with grief: and we hid as it were our faces from him; he was despised, and we esteemed him not. Surely he hath borne our griefs, and carried our sorrows: yet we did esteem him stricken, smitten of God, and afflicted. But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed. All we like sheep have gone astray; we have turned every one to his own way; and the LORD hath laid on him the iniquity of us all. He was oppressed, and he was afflicted, yet he opened not his mouth: he is brought as a lamb to the slaughter, and as a sheep before her shearers is dumb, so he openeth not his mouth. He was taken from prison and from judgment: and who shall declare his generation? for he was cut off out of the land of the living: for the transgression of my people was he stricken. And he made his grave with the wicked, and with the rich in his death; because he had done no violence, neither was any deceit in his mouth. Yet it pleased the LORD to bruise him; he hath put him to grief: when thou shalt make his soul an offering for sin, he shall see his seed, he shall prolong his days, and the pleasure of the LORD shall prosper in his hand. He shall see of the travail of his soul, and shall be satisfied: by his knowledge shall my righteous servant justify many; for he shall bear their iniquities. Therefore will I divide him a portion with the great, and he shall divide the spoil with the strong; because he hath poured out his soul unto death: and he was numbered with the transgressors; and he bare the sin of many, and made intercession for the transgressors.Christ was treated as though he was a sinner ("numbered with the transgressors") and specifically received this treatment from God ("it pleased the LORD to bruise him; he hath put him to grief") and particularly as a result of attributing our sins to him ("the LORD hath laid on him the iniquity of us all").
So, we agree with Bryan's first premise, namely that penal substitution requires Christ being punished by God. Moreover, we affirm that Scripture teaches this, something that Bryan (in this argument) does not dispute. One supposes that Bryan would dispute this point, but at least in the context of this argument he has not presented any exegetical reasons for doing so.
Instead, Bryan has attempted to argue that the conclusion conflicts with orthodox Christology and/or orthodox Trinitarian theology. He argues this by first asserting:
Punishment requires a loss of communion between God and Jesus?
Bryan's second premise is ambiguous. The term "loss of communion" can refer to a variety of different things. Bryan was asked a number of times to clarify what he meant by "communion" a number of times, but he declined to provide any clarification. We could reject Bryan's second premise on this ground alone. We don't need to accept premises that have undefined and ambiguous terms, particularly because such terms can lead to equivocation when it comes time to draw conclusions from them.
Nevertheless, we can answer this premise by distinguishing.
Punishment of Jesus by God does not require a loss of communion in the sense of God and Jesus being actually at odds. Jesus underwent the punishment of humiliation, including suffering and death, willingly. It is written: "Saying, 'Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done.'" (Luke 22:42) And again: "Who needeth not daily, as those high priests, to offer up sacrifice, first for his own sins, and then for the people's: for this he did once, when he offered up himself." (Hebrews 7:27) Had Christ been an unwilling victim, we might have said that the will of Christ and the will of God were at odds, but Christ submitted himself according to his human will to the will of God.
Thus, at a minimum, this premise is not true in every sense of the term "communion."
Bryan argued that punishment involves loss of communion in some sense, and that it is this loss of communion that primarily distinguishes punishment from discipline. Bryan is wrong. The primary distinction between punishment and discipline is the intent of the one inflicting the punishment or discipline.
In the case of punishment, the primary intent is to restore justice. In the case of discipline, the primary intent is to improve the disciplined person. It is worth noting that substitutionary punishment makes sense, while substitionary discipline largely does not. One is reminded of the prince's "whipping boy" in The Prince and the Pauper. While justice may be served by a man being flogged for a crime committed that merits flogging, in general the ill-behaving does not learn his lesson by another being flogged.
It is true that in the usual case, without substitution, there is typically an accompanying attitude of fundamental displeasure with the person being punished and an accompanying attitude of fundamental pleasure with the person being disciplined. Thus, a father beats a son whom he loves, although of course the father does not love the son's behavior that led to the need for the beating. If you are a modernist who thinks that beating children is immoral, read the Bible - but for the sake of this illustration just substitute "time out" for beating.
And ye have forgotten the exhortation which speaketh unto you as unto children, My son, despise not thou the chastening of the Lord, nor faint when thou art rebuked of him: for whom the Lord loveth he chasteneth, and scourgeth every son whom he receiveth. If ye endure chastening, God dealeth with you as with sons; for what son is he whom the father chasteneth not? But if ye be without chastisement, whereof all are partakers, then are ye bastards, and not sons. Furthermore we have had fathers of our flesh which corrected us, and we gave them reverence: shall we not much rather be in subjection unto the Father of spirits, and live? For they verily for a few days chastened us after their own pleasure; but he for our profit, that we might be partakers of his holiness. Now no chastening for the present seemeth to be joyous, but grievous: nevertheless afterward it yieldeth the peaceable fruit of righteousness unto them which are exercised thereby.Moreover, one can take the case of restitution as an example of retributive justice. Justice can be served by the victim of theft receiving treble restitution for his losses, but that justice is served regardless of the source of the funds. If those funds come from the criminal, they may have a disciplinary effect on the criminal, but even if they come from a substitute, they still make the injured person whole again.
Nevertheless, there is a sense in which communion, in the sense of felt favor of God, may have been lost. While we need not be dogmatic about it, it is possible for Christ, on the cross, to have lost a sense or awareness of the presence and favor of God. Christ was unaware, according to his humanity, of the day and the hour of the second coming. Likewise, it was possible for him to be unaware, according to his humanity, of the pleasure and favor of God toward him for a time on the cross.
Such an absence of awareness of God's presence and favor is one of the penalties that produce suffering for those in hell. Christ could undergo that same punishment in terms of suffering without actually losing God's presence or favor. Therefore, if this falls within the ambit of "communion" in the sense that Bryan means, Christ may have undergone it on the cross.
Loss of Communion with God Implies Some Heresy or Other?
Bryan's third premise depends heavily on the sense in which he means "communion," a sense he's seemingly unwilling to disclose. If Bryan is suggesting that punishment requires God the Father to stop loving the Son in every sense, then we simply disagree with Bryan's assertion. Suggesting that God the Father stopped loving the Son in every sense is clearly wrong.
Likewise, it is wrong to state that the Trinity was somehow severed by the cross. The intra-trinitarian communion was not damaged by the cross. Indeed, Christ was unified in will with the Father and the Spirit in the purpose of the crucifixion. If Christ and the Father were actually at odds, this would imply a serious error.
Furthermore, it is wrong to state that one person (Christ the God) was actually at odds with another person (Christ the Man). Christ is one person in two distinct natures. That means that Christ has two wills, but as one person Christ is unable to "commune" with himself, much less "lose" or "break" communion with himself.
On the other hand, Christ merely ceasing to be aware of God's presence or favor for a time on the cross according to his humanity does not imply any sort of heresy. So, much hinges on what Bryan means by "communion." Therefore, we cannot grant his third premise outright, just as we cannot grant his second premise outright. Instead, we need to distinguish in each case.
Update: In the comment box, Bryan Cross denies that he holds to the second premise. I've provided some documentation that seems to suggest he once advocated that premise. Nevertheless, he recently continued the argument in the comment box by alleging that the essence of hell punishment in particular is loss of communion with God. Even with this modification, the response above largely maintains. A few parts may not be relevant, but the rest is.