The answer to the trilemma is fairly trivial: While refusal to believe is indeed a sin, the reason it keeps one from salvation is not due to it being a sin, but because unbelief, by definition, precludes belief in Christ, without which no one can be saved (Heb 11:6). As with all sins Christ died for, forgiveness for unbelief is only obtained through subsequent belief in Him.

While Owen’s argument is trifling at best, his sophomoric reasoning isn’t the biggest problem here. While unstated, there is a premise both insidious and heretical that one must hold to make this argument without duplicity or cognitive dissonance: If one is seriously arguing that unbelief would not stop a person from being forgiven any more than any other sin, that is effectively saying that Christ’s death brings about salvation whether or not one believes. Or to put it succinctly,

If Christ died for one’s sins, then faith isn’t necessary for salvation.

Note that he is not arguing that all for whom Christ died must eventually believe and be saved, no, he is saying they would be saved despite not believing! Any Bible-believing Christian should be horrified by such a godless and contra-scriptural idea. That faith is absolutely necessary to be saved is all over the New Testament (in John 3:16, Acts 13:39, Romans 3:22, 5:1, 10:9, Galatians 2:16, 3:22, to give a few references). How in the world can an allegedly Christian theologian be arguing that lack of faith wouldn’t stop someone from receiving forgiveness?

https://arminianperspectives.wordpress.com/2019/09/27/the-cancer-in-calvinism/

To which I may add this dilemma to our Universalists:– God imposed his wrath due unto, and Christ underwent the pains of hell for,
either all the sins of all men,
or all the sins of some men,
or some sins of all men.
If the last, some sins of all men, then have all men some sins to answer for, and so shall no man be saved; for if God enter into judgment with us, though it were with all mankind for one sin, no flesh should be justified in his sight: “If the Lord should mark iniquities, who should stand?” Ps. cxxx. 3. We might all go to cast all that we have “to the moles and to the bats, to go into the clefts of the rocks, and into the tops of the ragged rocks, for fear of the Lord, and for the glory of his majesty,” Isa. ii. 20, 21.
If the second, that is it which we affirm, that Christ in their stead and room suffered for all the sins of all the elect in the world.
If the first, why, then, are not all freed from the punishment of all their sins?
You will say, “Because of their unbelief; they will not believe.”
But this unbelief, is it a sin or not? If not, why should they be punished for it?
If it be, then Christ underwent the punishment due to it, or not.
If so, then why must that hinder them more than their other sins for which he died from partaking of the fruit of his death?
If he did not, then did he not die for all their sins.
Let them choose which part they will.

(John Owen, The Death of Death in the Death of Christ, Book 1, ch 3; emphasis and spacing mine)

Owen’s argument doesn’t state or assume that people can be saved apart from faith or that they can’t be saved apart from faith. It is an argument in regards to how can individuals be atoned for but still punished. It rests on the contradiction of being technically covered by Christ(innocent) but yet still guilty of the punishment he takes for everyone at the cross. If a man is exonerated and his punishment placed on another, but then that punishment being placed back onto the exonerated because of the atoned deeds would require some basis for such an action on the judges part(unless he acts arbitrarily). It seems like the sacrifice of the other was pointless and the other man’s punishment seems unjust because these things were atoned for. This leaves God with about four options:

P1. Either all sin are atoned for and therefore everyone is saved.
P2. Some sins are atoned for and some are saved.
P3. All sins are atoned but only some are saved.
P4. All sins are atoned for and everyone is unsaved. 

Arminians affirm the third position. But what basis does God condemn people on? The fourth position has no chance of being true. We at least know some people are saved. The first is known to be false because we know some are damned and an Arminian won’t affirm the second position. So, the question arises that if Christ has taken care of the sins of the world(or the guilt of sin) then on what basis can God judge those that will end up in hell?

 

There’s been at least one Calvinist who argues that Owen isn’t assuming one can or can’t be saved apart from faith, and that he’s only asking what basis God can condemn a person if Christ has died for his sins.

It should already be evident that one cannot argue that a person would be saved in spite of unbelief without assuming that belief isn’t necessary to be saved. Though Owen phrases it as a question in his famous trilemma argument, he confirms elsewhere in the same work that this is what he actually believes:

“Fourthly, If this reconciliation of the world consist (as it doth) in a non-imputation of sin, then this is either of all their sins, or only of some sins. If of some only, then Christ saves only from some sins. If of all, then of unbelief also, or it is no sin; then all the men in the world must needs be saved, as whose unbelief is pardoned.” (John Owen, The Death of Death in the Death of Christ, Book 4, ch 3)

Notice that Owen’s argument is reductio. He argues that if you believe Christ dealt with sin/sin guilt or whatever makes mankind guilty before God, then it follows that all men must be saved. It never states the people will be saved apart from faith. It poses a dilemma for the Arminian where a man is innocent and guilty of the same crimes at the same time. But further just to show that it doesn’t presuppose that someone can be saved without faith. You can maintain that God created a world where postmortem repentance occurs to all, freely. Jerry Walls, the Arminian philosopher, maintains purgatory of some sort. So, it wouldn’t be hard for an Arminian to hold a strange position like that. Does Owen’s argument state that that is impossible? No. But you would have to be a crazed liberal the likes of Randal Rauser to believe that. But it assumes that the Arminian is against that position because they maintain some are damned. But how can their sins be atoned for and yet they are condemned for sin? At least, they are in a neutral state. They can’t be condemned but they didn’t believe so Jesus’ righteousness isn’t imputed to them. A better model is that Christ’s sacrifice and a person possessing faith are linked. God has designed the atonement by predestining it to be applied to the elect. So, Limited atonement makes more sense when considering the state of the damned(or better known by Arminians as ‘The Neutral’).