The early muslims couldn’t have appealed to the Bible even if they wanted to because, at the time of Muhammad’s ministry, the Bible had not yet been translated into arabic (and even then, the overwhelming majority of Makkan and Medinan people were illiterate).
I know you are not interested and sorry if this sounds like pushing facts down your throat: There are Christian scriptures in Arabic (not necessarily all canonical Bibles used by mainstream Christianity), primarily among which was the Infancy Gospel to the Arabs.
There are differing accounts of when the Infancy Gospel to the Arabs was written. Most non-Muslim scholars opine that it was the 5th century, meaning before Islam. Muslim scholars tend to conclude that it was translated into Arabic after Islam, so removed one possibility that Mohammad was influenced by an Arabic gospel. I am not sure whether any evidence has been presented to back up this assertion or is it another example of facts following conclusion (Mohammad couldn’t have been influenced by anyone else regarding the Quran and so the Arabic Gospel must have been translated after Mohammad).
The fact that the overwhelming majority of the people in Arabia at that time was illiterate cannot be a factor in whether an Arabic Bible exists. (Its like saying that most Jewish men in Jesus’ time was married and so Jesus must have been married; or that most American presidents have been white and so Obama couldn’t have been president). After all, most Christian evangelisation (at least in my part of the world) started with the translation of the catechism/Gospels/NT/Bible into the vernacular even though only the ruling elite was literate. In fact, some peoples did not have a written language and the translation of the Bible started with developing a written language where one did not exist - so you can have a vernacular Bible even if everyone is illiterate.
The influence of non-canonical Gospels on the Quran is very clear. The Quran contains two stories about Isa (Jesus to us) that are absent from our Bible (i) the baby Isa spoke in the cradle (sura 19:29-30) and (ii) the child Isa creating birds out of clay and breathed into them to make them fly (sura 5:110). Both these stories can be found in the Infancy Gospel to the Arabs - (i) v1 & (ii) v 46. The Infancy Gospel to the Arabs is a based on the Infancy Gospel of Thomas (different from the Gnostic Gospel of Thomas), probably written in late 2nd century, which was later translated into Syriac and from there into Arabic.
The Quran also have a very exhaustive treatment of Maryam (Mary to us), with Maryam being mentioned more often in the Quran than Mary in the Bible. Indeed, she even have sura 19 named after her. Again much of the material came from the Protoevangelium of James. (Before Catholics start discounting this apocryphal book, this book has provided us with the names of the parents of Mary and many Catholics would recognise stories in it that resembles our Marialogical stories). it was known to be translated into Arabic but I am not aware if any manuscript survived.
Add to it the role of Waraqa bin Neufal (uncle to Mohammad’s wife, to whom he turned after being confused after his first encounter with the angel Jibril), whose Ebionite (I assume that his is an Ebionite despite a minority & Muslim view that he was Nestorian) views correspond with that of Mohammad - Ebionites deny the divinity of Jesus (that Jesus was only a prophet), the crucifixion & resurrection (Jesus did not die on the cross) and the Trinity (the prophets did not speak with the Holy Spirit), coupled with great devotion to the Mosaic Law. Ebionites were believed have survived into the 12th century in south-western Arabia with one Muslim account from that time referring to Jews who accept Jesus as a prophet.
Ebionite scriptures include the OT books plus only an amended Gospel of Matthew (minus the first 2 chapters), sometimes referred to as the Gospel to the Hebrews. If Waraqa was an Ebionite, Mohammad would have been sympathetic with the view that there is only one Gospel and that Christians have added/amended to that Gospel.
Besides these, there were many Christian Arabs tribes during the time of Mohammad, including Catholic-Orthodox Christians (St Aretas was a Arab martyr). Arabs have been Christians since NT times, with Arabs being among those preached to on Pentecost day (Acts 2:11) and Paul referred to a journey he made to Arabia (Gal 1:17). Those Arab Christians must have had scriptures and those who are not mainstream Christians would have their own apocryphal scriptures, not to mention non-Christian sects like Mandeans who have scriptures with at least a passing mention to Jewish & Christian figures. How many of these scriptures are known to Mohammad and had influenced him (I find) is hard to say. I find much of the theories speculative and circumstantial.
So for now, I would stick to the Infancy Gospel of the Arabs (or Thomas), the Protoevangelium of James and the Gospel of the Hebrews being known to Mohammad that could have shaped his thinking about Christianity. Would be happy for anyone to add to this with some scholarly evidence - no speculation please.