Additionally, it does not make sense to directly compare the process of compilation of the Bible and the Qur'an because the two scriptures are quite different structurally and in terms of the historical circumstances surrounding them.
The Qur'an is much newer than the Bible, much shorter than the Bible, emerged in one region and in a short span of time. There has never been any question of what portions of the Qur'an to consider canonical. It was also all in a single language and dialect. Stylistically, the Qur'an is internally consistent indicating it is the work of a single author and does not have interpolations (whether or not one considers that author to be God, it still has a single "voice"). While, at the time of the Prophet, Mecca and Medina did not have sophisticated means of producing books, it was not long before advances in those areas, such as the use of paper, came to the Islamic world; and, in fact, it was through the Islamic world that some of these technologies were transferred to Europe.
In contrast, the compilation of the Bible is much more complicated. The scriptures are much older, span a broader length of time, and there is a lot more complexity with respect to how it came to be a single text. (There are numerous works on this subject available.) It becomes even more complicated if one considers both the New and the Old Testaments. It is not as if someone 2000+ years ago held out a book (or, rather, a scroll) called "The Bible" that is exactly the same as you find today; there was a process.
So, from a purely historical perspective, there are very different circumstances surrounding the compilation of what today we know as the Bible and the Qur'an, and the nature of the two books is also quite different. Therefore, it is not a one-to-one comparison and it is best to look at the compilation of each scripture separately and consider the validity or authenticity of each one separately.
Quran is the only Holy text which is completely intact, unlike Torah and Bible which have been distorted and changed into many different versions.
There is possibility at of misquoting the prophet Muhammad (SAWA) while writing the Quran as the Prophet and the writer (Imam Ali) both are Infallible supported by Allah (SWT) and His Angles. Allah (SWT) has promised to keep the Quran intact ( Verily, We Have sent the Thikr (Quran) and We will be guarding it (from corruption))(انا نحن نزلنا الذكر وانا له لحافظون )(Sura 15, verse 9)
The Prophet (SAWA) has recited the Quran and supervised writing it. No question at all of any error in writing the Quran as all the Infallible Imams read the Quran in the same way which confirms that there is no error whatsoever.
The fact of the purity of Quran is admitted even by the enemies who tried every thing to harm Quran but could not as Quran is protected by Allah (SW) and no one or power can make any change in Quran as it's enemies tried and failed.
Thank you for the interesting question!
As you know, there are a number of differences between the Qur'anic and Biblical stories of similar people. Muslims and those who believe in a literal interpretation of the Old Testament will have to agree to disagree until the day comes when the truth is made clear!
In this case, it is more complicated because it is not 100% clear that Talut is the same person as Saul because they have different names.
There is a good discussion of the story in the relevant section of The Study Quran, ed. S. H. Nasr. Also here are some hadith on the subject. https://www.al-islam.org/hayat-al-qulub-vol-1-allamah-muhammad-baqir-al-majlisi/account-ishmaeil-talut-and-jalut
By the nature of oral literature, when a story is told, it is often altered or embellished, and this is not considered a problem in cultures that transmit their literature and history orally. The oral retelling of historical events still continues in many parts of Africa, almost as a spiritual ritual. The genealogist, or story-teller, will claim to have heard the story from his father, who heard it from his father, and so on. There is meant to be a chain of transmission which should protect against error; nevertheless, while you can find the same basic framework of the narrative in different accounts, there will often be notable differences.
The narratives of the Bible were compiled over time by different groups of scribes. Initially, narratives from the Bible were transmitted orally - even for several generations. Eventually they came to be written down. The accounts that have been written in the Bible may not be the only accounts that exist of certain narratives. There may be several variations.
In fact, often certain points that the Qur'an makes about particular doctrines held among Jews or Christians actually refer to apocryphal beliefs that were not held by the majority. There were often sects and sub-sects in Arabia and Iraq and not much has been written about them.
The Holy Quran describes the story in Chapter 2, Surah Baqarah, from verse 246 to 251. We believe the Qur'an is the word of God which is unchanged and unfabricated and 100% true. However the bible has gone through much distortion and fabrication. Therefore the story of Talut in the Quran is the accurate version.
There are historic references in hadith litrature but scholars are careful in taking these as they have to match the Quran.
Here is a lecture on Talut that I gave that you may benefit from: