The Quran suggests Muhammad is written about in what we have of the Torah and the Injeel in Quran 7:157 and other verses and hadiths.
Quran 7:157 "Those who follow the Messenger, the unlettered prophet, whom they find written in what they have of the Torah and the Gospel..."
One of the prooftexts often provided is Deuteronomy 18:15-20.
Deuteronomy 18:15-20 [Moses speaking to the Israelites]15 The Lord your God will raise up for you a prophet like me from among your midst, from your brothers. You must listen to him. 16 For this is what you asked of the Lord your God at Horeb on the day of the assembly when you said, “Let us not hear the voice of the Lord our God nor see this great fire anymore, or we will die.”
17 The Lord said to me [Moses]: “What they say is good. 18 I will raise up for them a prophet like you from among their brothers, and I will put my words in his mouth. He will tell them everything I command him. 19 I myself will call to account anyone who does not listen to my words that the prophet speaks in my name. 20 But a prophet who presumes to speak in my name anything I have not commanded, or a prophet who speaks in the name of other gods, is to be put to death.”
What is the Islamic basis for understanding that a prophet "among your midst, among your brothers" could include the Ishmaelites as a possibility that would fulfill these verses?
An answer much more thorough than "because it doesn't exclude them explicitly" would be appreciated, as this is counter-intuitive. Additionally, the following data should be taken into consideration:
In the same speech of Moses speaking to the Israelites, "brothers" seems to be more definitely restricted to Israelites (Deuteronomy 18:1-2), as the "brothers" have an inheritance, yet the Ishmaelites have no mention of an "inheritance" in Scripture. Again, in Deuteronomy 17:14-15, "brothers" are only part of those entering the Promised Land, the "land that God is giving you," which the Ishmaelites had no place in. The Ishmaelites were geographically quite separated from the Israelite location, so "among your midst" additionally appears to, on the surface, restrict this prophet to be an Israelite. Finally, "brothers" as a familial term seems to also restrict the prophet to be an Israelite. The Israelites and Ishmaelites, at this point, had already been separated in geography and genealogy for over 400 years; while it may seem reasonable to apply the phrase to relatively close genealogical members (though this distance isn't applied in chapters 17-18 mentioned previously), it seems highly unusual to use this term for those separated by such a great blood-relationship distance, many generations ago.
So, really, to answer the main question as well as to address the all the data necessary, sub-questions would have to be answered (or counter-examples given):
What good reason do Muslims propose is there to change the scope of the phrase "brothers" to include the Ishmaelites when it is previously restricted to Israelites in 17:14-15 and 18:1-2?
What good reason do Muslims propose is there for "among your midst" to apply to a people group that seems to be not "among your midst"? (e.g. are there counter-examples in the Tanakh where "among your midst" also applies to two things geographically separated by many miles?)
I would really appreciate direction toward any Muslim scholarly literature that exists on the subject as well as any Muslim scholars familiar with Biblical Hebrew. A discussion of the use of the term "brothers" and "among your midst" specifically in the Tanakh seems warranted. I recall "brothers" is used at some point in Scripture, though not in the same context, of Esau's descendants, which would still be Isaac's descendant. I welcome answers that dispute with any of the claims throughout this question. Let's leave the proposed corruption of the Bible out of the scope of this question.