If we don't have the original biblical documents, but only possess copies (that have copyist errors), then how can we argue that the original documents were without error?
We don't need to have the original documents in order to demonstrate their inerrancy any more than a prosecutor needs a body to prove a crime has been committed. Inferences can be drawn from the evidence at hand, and a reasonable conclusion can be argued from biblical principles.
First of all, the copies definitely have errors in them, but whether or not they have errors does not necessitate that the originals did. We don't need to have the original documents in order to demonstrate their inerrancy anymore than a prosecutor needs a body to prove a crime has been committed.
Second, the Scriptures are said to be God-breathed, i.e., inspired.
Second Timothy 3:16 says,
"All Scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness."
The word "inspired" is 'theopneustos', God-breathed.
This means that Scripture comes from God. Jesus said in Luke 24:44-45,
"These are My words which I spoke to you while I was still with you, that all things which are written about Me in the Law of Moses and the Prophets and the Psalms must be fulfilled." 45 Then He opened their minds to understand the Scriptures."
Notice that the Scriptures are referred to as the entire Old Testament. Furthermore, Jesus said in John 10:35,
“If he called them gods, to whom the word of God came (and the Scripture cannot be broken)."
We see that Jesus speak for the entire Old Testament, to which John the apostle refers to Scripture. Then we see that Jesus said the Scripture cannot be broken. This means that it cannot fail. From this we can easily conclude that the position of Jesus is that the Scriptures, the written form, cannot fail, cannot be broken. Logically, this means that they are without error.
Third, Jesus did not deal with the issue of copyist errors. He simply stated that God's word cannot be broken. Likewise, we should take the same stance.
Fourth, Jesus did not possess the original writings, yet he said that they could not be broken and contain error.
Fifth, the New Testament is likewise considered Scripture by the Christian Church. Therefore, it falls under the purview of all Scripture being inspired of God.