Whether or not you find his position plausible, the argument seems unusually defensive. And I think he's aware of the wobbliness... An oddly atypical bit of Barth.

CD I/2, p.529-530:

...every time we turn the Word of God into an infallible biblical word of man or the biblical word of man into an infallible Word of God we resist that which we ought never to resist, i.e., the truth of the miracle that here fallible men speak the Word of God in fallible human words - and we therefore resist the sovereignty of grace, in which God Himself became man in Christ, to glorify Himself in his humanity. If we cannot make up our minds for this hard thinking, let us see to it that we are not shutting ourselves off from the real word of comfort spoken to us by the existence of the Bible as such. And if we want to assert a supposedly stricter concept of the value and authority of the Bible, let us see to it that we are not moving away from the strictness of its true value and authority. If the prophets and apostles are not real and therefore fallible men, even in their office, even when they speak and write of God's revelation, then it is not a miracle that they speak the Word of God... To the bold postulate, that if their word is to be the Word of God they must be inerrant in every word, we oppose the even bolder assertion, that according to the scriptural witness about man, which applies to them too, they can be at fault in any word, and have been at fault in every word, and yet according to the same scriptural witness, being justified and sanctified by grace alone, they have still spoken the Word of God in their fallible and erring human word. It is the fact that in the Bible we can take part in this real miracle, the miracle of the grace of God to sinners, and not the idle miracle of human words which were not really human words at all, which is the foundation of the dignity and authority of the Bible.

And then, p.531:

We are absolved from differentiating the Word of God in the Bible from other contents, infallible portions and expressions from the erroneous ones, the infallible from the fallible, and from imagining that by means of such discoveries we can create for ourselves encounters with the genuine Word of God in the Bible. If God was not ashamed of the fallibility of all the human words of the Bible, of their historical and scientific inaccuracies, their theological contradictions, the uncertainty of their tradition, and above all, their Judaism, but adopted and made use of these expressions in all their fallibility, we do not need to be ashamed when He wills to renew it to us in all its fallibility as witness, and it is mere self-will and disobedience to try to find some infallible elements in the Bible.

TL;DR? here's what Barth asserts: don't connect the dots. Barth, CD I/2, p.483-484:

Rightly understood, the unity of Holy Scripture gives rise to a conclusion & demand to which the Church must pay good heed. But this conclusion & demand is not that we should abstract from the Bible some concealed historical or conceptual system, an economy of salvation or a Christian view of things. There can be no biblical theology in this sense, either of the Old or New Testament, or of the Bible as a whole. The presupposition & organising centre of such a system would have to be the object of the biblical witness, that is, revelation. Now revelation is no more & no less than the life of God Himself turned to us, the Word of God coming to us by the Holy Spirit, Jesus Christ. But in our thinking, even in our meditation on the biblical texts, it is only improperly, i.e., only in the form of our recollection & expectation, that we can "presuppose" Jesus Christ & then add to this presupposition other thoughts, even those which are derived from our exposition of those texts. Properly, & that means, in living fact, revelation can only be presupposed to our thoughts, even to those based on exposition, that is, it can only be their organising centre by revelation itself. Therefore, a biblical theology can never consist in more than a series of attempted approximations, a collection of individual exegeses. There can never be any question of a system in the sense of Platonic, Aristotelian or Hegelian philosophy... Even the biblical witnesses themselves cannot & do not try to introduce revelation of themselves. They show themselves to be genuine witnesses of it by the fact that they only speak of it by looking forward to it & by looking back at it. How can we wish to complete the totality of their witness by treating revelation as a presupposition which we can control? How can we expound it except by surrendering ourselves with them to the recollection, their recollection, and to the expectation, their expectation? It is only in this surrender - and not in an arbitrary doing of what they omitted to do - that our exposition of that witness will be kept pure and will become our own witness. Biblical theology (and self-evidently dogmatics too) can consist only in an exercise of this surrender, not in an attempt to introduce the totality of the biblical witness.

At this point we must ask whether the older Protestant theology of the 17th century did not do too much, and therefore too little. Intrinsically, there can be no objection to the fact that in its exposition it made such active use of the instruments of Aristotelian and later Cartesian philosophy. How can we find fault, and not take as a model, the comprehensive thoroughness & accuracy which it obviously sought & in such surprising measure revealed? If only it had kept itself freer from the temptation to be inspired to go further & to seek that which is theologically impossible, a systematics of revelation, a system in which revelation itself can be used as a presupposition! It attempted to bring in the witness of revelation as such in its unity & entirety. But in so doing it did violence to it. And it was on this that it foundered when the Philistines came upon it in the 18th century as once they had come upon Samson. We must leave it to revelation itself to introduce itself either in its unity & entirety or indeed at all. Revelation is never behind us: always we can only follow it. We cannot think it: we can only contemplate it. We cannot assert & prove it: we can only believe it, believe it in recollection & expectation, so that if our faith is right and well-pleasing to God in what we then think & say, it can assert & prove itself.