When we say that we believe in the inspiration of the original manuscripts, this in no way implies that we mean we do not have an inspired and inerrant Bible. We believe that we do. We have in our Greek texts, whether one uses the majority text or the Nestle Aland text, etc., a text that is accurate and gives us God’s Holy Word. This would also apply to the original Hebrew manuscripts (MSS).
Having said that, I believe we must also face facts. Sometimes, because of our desire for certainty, people seem to be more interested in the pursuit of certainty than in the pursuit of truth. The facts are that we do not have the original manuscripts. Instead, we have thousands of manuscripts, and no two of them are exactly the same. Even those who advocate the text of the Textus Receptus (TR) must fact the reality that no two of the MSS making up the TR are precisely alike, defying the claims that pure uniformity can be discovered. We must also face the fact that there were other English translations before the 1611 KJV, which itself was revised in 1769. The early, fifth-century Latin Vulgate was consulted by King James translators.
Scripture states God’s promise to keep his Word pure, which we believe applies to translations conducted by godly scholars in every age. The original is what is inspired and inerrant, and insofar as a translation accurately represents that original, then we have an inspired, inerrant Bible. Those who believe that the KJV is the preserved and inspired Bible ultimately end up crediting to fallen human copyists the same degree of inspiration, by way of preservation, as was afforded to the inspiration of the human authors of Scripture themselves.
Because of textual criticism with scholars using the thousands of manuscripts we have at our disposal, we know that our Bible is tremendously accurate with the variations in the manuscripts being less than one percent (if I remember correctly), and these variations do not involve any major doctrine. This means we have an inspired and inerrant Bible, but it also means that we can thank God for textual scholars who labor intensely to compare the thousands of manuscripts in order to take us as close to the original as possible, and I believe, in God’s providence, He has done just that. In fact, one of the marvels of the Bible and evidences of its uniqueness and God’s preservation, in comparison to other ancient writings, is the number of manuscripts (some 5000), but these MSS, though tremendously similar, do vary.
There is much more that could be said about this, but I hope this answers your question about the position of BSF, which, by the way, is the position of a large majority of Bible-loving scholars in the Christian world today.