The relationship between Jesus' humanity and divinity is paradigmatic of but different in kind from any other relation between God and humanity. It is paradigmatic for this relation because it shows us how God relates to us; and it is different in kind because Jesus is the eternal Son incarnate.
Out of the question is anything that would make the biblical text another incarnation in any sense. So, any stability that we seek between the work of the Spirit and the biblical text must be different in kind than the relationship between God and humanity in the incarnation, which means that it is another case of that relationship paradigmatically demonstrated in the incarnation.
What is that paradigm? Conceptually speaking, Chalcedon: a unity in distinction of divine and human. Outside the incarnation, however, the fully human part is always true but the fully divine part never is. Instead, we have something like "fully human, use instrumentally by God."
Now, there can be degrees of this instrumental use. The special authority and stability of Scripture, which places it beyond the dead dog, is tied to the relation of God's activity to it.
This activity proceeds, accompanies and follows. It proceeds in that Jesus grants teaching authority to the disciples (NT at least, you need an account of prophets for the OT), teaches them, etc. It accompanies in the Spirit's inspirational and guiding work as the authors write. It follows in that the Spirit employs what the authors produced as an instrument to quicken us.
Now, it could very well be true - and in a sense it is - that the Spirit acts in analogous ways with our dead dog. But, what the dead dog lacks is Jesus' commission. Thus, it is on account of the authority granted to the apostles - ratified by the continual witness of the Spirit to the biblical text; what Calvin called Scripture's self-authentication - that it must have pride of place, and indeed sole normative status (Barth is a Protestant, after all) for church and Christians.
That is how I think Barth's logic goes.